LIBRARY 


AN     ELEMENTAKY     TREATISE 


FOURIER'S   SERIES 


SPHERICAL,  CYLII^DHIOAL,  AND  ELLIPSOIDAL 
HARMONICS, 


APPLICATIONS  TO  PROBLEMS  IN  MATHEMATICAL  PHYSICS. 


WILLIAM   ELWOOD   BYERLY,    Ph.D., 

PROFE.SSOK    OF    MATHEMATICS    IN    HARVARD    UNIVERSITY. 


BOSTON,  U.S.A.: 
GINN    &   COMPANY,    PUBLISHERS, 

1895. 


Copyright,  1893, 
Bv  WU.IJAIVt  JELWOOD  BTERLY. 


'all  rights 


KESEKVED. 


A-C-^X*'^-^''^ 


itr.H- 


PREFACE 


About  ten  years  ago  I  gave  a  course  of  lectures  on  Trigonometric  Series, 
following  closely  the  treatment  of  that  subject  in  Riemann's  "Partielle 
Differentialgleichungen,"  to  accompany  a  short  course  on  The  Potential 
Function,  given  by  Professor  B.  0.  Peirce. 

My  course  has  been  gradually  modified  and  extended  until  it  has  become  an 
introduction  to  Spherical  Harmonics  and  Bessel's  and  Lame's  Functions. 

Two  years  ago  my  lecture  notes  were  lithographed  by  my  class  for  their 
own  use  and  were  found  so  convenient  that  I  have  prepared  them  for 
publication,  hoping  that  they  may  prove  useful  to  others  as  well  as  to  my 
own  students.  ]Meanwhile,  Professor  Peirce  has  published  his  lectures  on 
"  The  Xewtonian  Potential  Function "  (Boston,  Ginn  &  Co.),  and  the  two 
sets  of  lectures  form  a  course  (jNIath.  10)  given  regularly  at  Harvard,  and 
intended  as  a  partial  introduction  to  modern  Mathematical  Physics. 

Students  taking  this  course  are  supposed  to  be  familiar  with  so  much  of  the 
infinitesimal  calculus  as  is  contained  in  my  "  Differential  Calculus  "  (Boston, 
Ginn  &  Co.)  and  my  "  Integral  Calculus  "  (second  edition,  same  publishers), 
to  which  I  refer  in  the  present  book  as  <'  Dif.  Cal."  and  "  Int.  Cal."  Here, 
as  in  the  "  Calculus,"  I  speak  of  a  "  derivative  "  rather  than  a  "  differential 

coefficient,"  and  use  the  notation  D^  instead  of  —  for  "  partial  derivative  with 

respect  to  cc." 

The  course  was  at  first,  as  I  have  said,  an  exposition  of  Riemann's  "Partielle 
Differentialgleichungen."  In  extending  it,  I  drew  lai'gely  from  Ferrer's 
"  Spherical  Harmonics  "  and  Heine's  "  Kugelfunctionen,"  and  was  somewhat 
indebted  to  Todhunter  ("Functions  of  Laplace,  Bessel,  and  Lame"),  Lord 
Rayleigh  ("Theory  of  Sound"),  and  Forsyth  ("Differential  Equations  "). 

In  preparing  the  notes  for  publication,  I  have  been  greatly  aided  by  the 
criticisms  and  suggestions  of  my  colleagues.  Professor  B.  0.  Peirce  and  Dr. 
Maxime  Bocher,  and  the  latter  has  kindly  contributed  the  brief  historical 
sketch  contained  in  Chapter  IX. 

W.  E.  BYERLY. 

Cambridge,  Mass.,  Sept.   1893. 

981582 


ANALYTICAL  TABLE  OF  CONTENTS. 


CHAPTER    I. 

PAGES 

Intkoimctiox 1-29 

Airr.  1.  List  of  some  important  homogeneous  linear  partial  differential  equations 
of  Physics.  — Auts.  2-4.  Distinction  between  the  (jeneral  solution  and  a  particular 
solution  of  a  differential  equation.  Need  of  additional  data  to  make  the  solution 
of  a  differential  equation  determinate.  Definition  of  linear  and  of  linear  and 
homogeneous.  — Arts.  5-(3.  Particttlar  solutions  of  homogeneotis  linear  differential 
equations  may  be  combined  into  a  more  general  solution.  Need  of  development 
in  terms  of  normal  forms.  —Art.  7.  Problem:  Permanent  state  of  temperatures 
in  a  thin  rectangular  plate.  Need  of  a  development  in  sine  series.  Example.  — 
Art.  8.  Problem:  Transverse  vibrations  of  a  stretched  elastic  string.  A  develop- 
ment in  sine  ser/es  stiggested. —  Art.  9.  Problem:  Potential  function  due  to  the 
attraction  of  a  circular  ring  of  small  cross-section.  Surface  Zonal  Harmonics 
(Legendre's  Coefficients).  Example. — Art.  10.  Problem:  Permanent  state  of 
temperatures  in  a  solid  sphere.  Development  in  terms  of  Surface  Zonal  Har- 
monics suggested. —  Arts.  11-12.  Problem:  Vibrations  of  a  circular  drumhead. 
Cylindrical  Harmonics  (Be.ssers  Functions).  Recapitidation.  —  Art.  13.  Method 
of  making  the  solution  of  a  linear  partial  differential  equation  depend  upon  solving 
a  set  of  ordinary  differential  eqtiations  by  assttming  the  dependent  variable  equal 
to  a  product  of  factors  each  of  which  involves  but  one  of  the  independent  variables. 
Arts.  14-1-5.  Method  of  solving  ordinary  homogeneotts  linear  differential  equa- 
tions by  develoiDment  in  power  series.  Applications.  —  Art.  Ki.  Application  to 
Legendre's  Equation.  Several  forms  of  general  solution  obtained.  Zonal 
Harmonics  of  the  second  kind. — Art.  17.  Application  to  Bessel's  Equation. 
General  soltttion  obtained  for  the  case  where  ?;;  is  not  an  integer,  and  for  the  case 
where  m  is  zero.  Bessel's  Function  of  the  second  kind  and  zeroth  order.  — Art. 
18.  Method  of  obtaining  the  general  sohttion  of  an  ordinary  linear  differential 
equation  of  the  second  order  from  a  given  particular  soltttion.  Application  to 
the  equations  considered  in  Arts.  14-17. 

CHAPTER    IL 
Development  in  Trigonometric  Series .'30-54 

Arts.  19-22.  Determination  of  the  coefficients  of  n  terms  of  a  sine  series  so  that 
the  sum  of  the  terms  shall  be  equal  to  a  given  fttnction  of  x  for  n  given  values 
of  X.  Numerical  example.  —  Art.  23.  Problem  of  development  in  sine  series 
treated  as  a  limiting  case  of  the  problem  jtist  solved. — Arts.  24-25.  Shorter 
method  of  solving  the  problem  of  development  in  series  involving  sines  of  whole 
mitltiples  of  the  variable.    Working  rttle  deduced.    Recapitulation. — Art.  26.  A 


VI  TABLE    OF    CONTENTS. 

PACKS 

few  important  sine  developments  obtained.  Examples.  — Abt«.  27-28.  Develop- 
ment in  cosine  series.  Examples.  — Art.  29.  Sine  series  an  odd  function  of  the 
variable,  cosine  series  an  even  function,  and  both  series  periodic  functions.  — 
Art.  30.  Development  in  .series  involving  both  sines  and  cosines  of  whole 
multiples  of  the  variable.  Fourier's  series.  Examples.  —  Art.  .31.  Extension  of 
the  range  witliin  which  the  function  and  the  series  are  equal.  Examples.  — 
Art.  32.    Fourier's  Integral  obtained. 

CHAPTER    III. 
Convergence  or  Fourier's  Series -55-68 

Arts.  3.3-36.  The  question  of  the  convergence  of  the  sine  series  for  unity  con- 
sidered at  length.  —  Arts.  37-38.  Statement  of  the  conditions  which  are  sutiicient 
to  warrant  the  development  of  a  function  into  a  Fourier's  series.  Hi.storical  note. 
Art.  39.  Graphical  representation  of  successive  approximations  to  a  sine  series. 
Properties  of  a  Fourier's  series  infen-ed  from  the  constructions. — Arts.  40-42. 
Investigation  of  the  conditions  under  whicli  a  Fourier's  series  can  be  differentiated 
term  by  term.  —  Art.  43.  Conditions  under  which  a  function  can  be  expressed  as 
a  Fourier's  Integral. 

CHAPTER    IV. 

Solution   of   Problems   in    Physics    i$v    the    Aid   of   Fourier's    Integrals    and 

Fourier's  Series 69-134 

Arts.  44-48.  Logarithmic  Potential.  Flow  of  electricity  in  an  infinite  plane, 
where  the  value  of  the  Potential  Function  is  given  along  an  infinite  straight  line; 
along  two  mutually  perpendicular  straight  lines;  along  two  parallel  straight  lines. 
Examples.  Use  of  Conjugate  Functions.  Sources  and  Sinks.  Equipotential 
lines  and  lines  of  Flow.  Examples. — Arts.  49-52.  One-dimensiunal  flow  of  heat. 
Flow  of  heat  in  an  infinite  solid;  in  a  solid  with  one  plane  face  at  the  temperature 
zero;  in  a  solid  with  one  plane  face  whose  temperature  is  a  function  of  the  time 
(Riemann's  solution);  in  a  bar  of  small  cro.ss  section  from  whose  surface  heat 
escapes  into  air  at  temperature  zero.  Limiting  state  approached  when  the  tem- 
perature of  the  origin  is  a  periodic  function  of  the  time.  Examples.  — Arts.  53- 
54.  Temperatures  due  to  instantaneous  and  to  permanent  heat  sources  and  sinks, 
and  to  heat  doublets.  Examples.  Application  to  the  case  where  there  is 
leakage.  —  Arts.  55-56.  Transmission  of  a  disturbance  along  an  infinite  stretched 
elastic  string.  Examples.  —  Arts.  57-58.  Stationary  temperatures  in  a  long 
rectangular  plate.  Temperature  of  the  base  unity.  Summation  of  a  Trigono- 
metric series.  Isothermal  lines  and  lines  of  flow.  Examples. — Art. -59.  Potential 
Function  given  along  the  perimeter  of  a  rectangle.  Examples.  —  Arts.  60-63. 
One-dimensional  flow  of  heat  in  a  slab  with  parallel  plane  faces.  Both  faces  at 
temperature  zero.  Both  faces  adiathermanous.  Temperature  of  one  face  a 
function  of  the  time.  Examples.  —  Art.  64.  INIotion  of  a  stretched  ela.stic  string 
fastened  at  the  ends.  Steady  vibration.  Nodes.  Examples.  —  Art.  65.  Motion 
of  a  string  in  a  resisting  medium.  —  Art.  66.  Flow  of  heat  in  a  .sphere  whose 
surface  is  kept  at  a  constant  temperature.  —  Arts.  67-68.  Cooling  of  a  sphere  in 
air.  Surface  condition  given  by  a  differential  equation.  Development  in  a  Trigo- 
nometric series  of  which  Fourier's  Sine  Series  is  a  special  case.     Examples.  — 


TABLE    OF    CONTENTS. 


■AGES 


Arts.  69-70.  Flow  of  heat  in  an  infinite  solid  witli  one  plane  face  which  is 
exposed  to  air  whose  teiuperatuie  is  a  function  of  the  time.  Solution  for  an 
instantaneous  heat  source  when  the  temperature  of  the  air  is  zero.  Examples.  — 
Arts.  71-73.  Vibration  of  a  rectangular  drumhead.  Development  of  a  function 
of  two  variables  in  a  double  Fourier's  Series.  Examples.  Nodal  lines  in  a 
rectangular  drumhead.     Nodal  lines  in  a  square  drumhead. 

Miscellaneous  Prohlems 135-148 

I.  Logarithmic  Potential.  Polar  Coordinates.  —  II.  Potential  Function  in  Space. 
III.    Conduction  of  heat  in  a  plane.  —  IV.    Conduction  of  heat  in  Space. 

CHAPTER   V. 

Zonal   Harmonics 144-194 

Art.  74.  Recapitulation.  Surface  Zonal  Harmonics  (Legendrians).  Zonal  Har- 
monics of  the  second  kind.  — Arts.  75-70.  Legendrians  as  coefficients  in  a  Power 
Series.  Special  values. — Art.  77.  Summary  of  the  properties  of  a  Legendrian. 
List  of  the  first  eight  Legendrians.  Relation  connecting  any  three  successive 
Legendrians.  —  Arts.  78-81.  Problems  in  Potential.  Potential  Function  due  to 
the  attraction  of  a  material  circular  ring  of  small  cross  section.  Potential  Function 
due  to  a  charge  of  electricity  placed  on  a  thin  circular  disc.  Examples:  Spheroidal 
conductors.  Potential  Function  due  to  the  attraction  of  a  material  homogeneous 
circular  disc.  Examples  :  Homogeneous  hemisphere  ;  Heterogeneous  sphere  ; 
Homogeneous  spheroids.  Generalisation. — Art.  82.  Legendrian  as  a  sum  of 
cosines. — Arts.  83-84.  Legendrian  as  the  ??ith  derivative  of  the  vith  power  of 
a;'^  — 1.  —  Art.  85.  Equations  derivable  from  Legendre's  Equation. — Art.  86. 
Legendrian  as  a  Partial  Derivative.  —  Art.  87.  Legendrian  as  a  Definite  Integral. 
Arts.  88-90.  Development  in  Zonal  Harmonic  Series.  Integral  of  the  product  of 
two  Legendrians  of  different  degrees.  Integral  of  the  square  of  a  Legendrian. 
Formulas  for  the  coefficients  of  the  series.  —  Arts.  91-92.  Integral  of  the  product 
of  two  Legendrians  obtained  by  the  aid  of  Legendre's  Equation;  by  the  aid  of 
Green's  Theorem.  Additional  formulas  for  integration.  Examples.  —  Arts.  93- 
94.  Problems  in  Potential  where  the  value  of  the  Potential  Function  is  given  on  a 
spherical  surface  and  has  circular  symmetry  about  a  diameter.  Examples.  — 
Art.  95.  Development  of  a  power  of  x  in  Zonal  Harmonic  Series.  —  Art.  96. 
Useful  formulas.  —  Art.  97.  Development  of  sin  n6  and  cos  nd  in  Zonal  Harmonic 
Series.  Examples.  Graphical  representation  of  the  first  seven  Surface  Zonal 
Harmonics.  Construction  of  successive  approximations  to  Zonal  Harmonic  Series. 
Arts.  98-99.  Method  of  dealing  with  problems  in  Potential  when  the  density  is 
given.  Examples. — Art.  100.  Surface  Zonal  Harmonics  of  the  second  kind. 
Examples:  Conal  Harmonics. 

CHAPTER   VL 

Spherical  Harmonics 195-218 

Arts.  101-102.  Particular  Solutions  of  Laplace's  Equation  obtained.  Associated 
Functions.  Tesseral  Harmonics.  Surface  Spherical  Harmonics.  Solid  Spherical 
Harmonics.  Table  of  Associated  Functions.  Examples.  —  Arts.  103-108.  De- 
velopment in  Spherical  Harmonic  Series.     The  integral  of  the  product  of  two 


Viii  TABLK    OF    CONTENTS. 

PAGES 

Surface  Spherical  Harmonics  of  different  degrees  taken  over  the  surface  of  the 
unit  sphere  is  zero.  Examples.  The  integral  of  the  product  of  two  Associated 
Functions  of  the  same  order.  Formulas  for  the  coefficients  of  the  series.  Illustra- 
tive example.  Examples. — Arts.  109-110.  Any  homogeneous  rational  integral 
Algebraic  function  of  x,  ?/,  and  z  which  satisfies  Laplace's  Equation  is  a  Solid 
Spherical  Harmonic.  Examples.  — Art.  111.  A  transformation  of  axes  to  a  new 
set  having  the  same  origin  will  change  a  Surface  Spherical  Harmonic  into  another 
of  the  same  degree.  —  Arts.  112-114.  Lcq^laclans.  Integral  of  the  product  of  a 
Surface  Spherical  Harmonic  by  a  Laplacian  of  the  same  degree.  Development  in 
Spherical  Harmonic  Series  by  the  aid  of  Laplacians.  Table  of  Laplacians.  Ex- 
ample.—  Art.  115.  Solution  of  problems  in  Potential  by  direct  integration. 
Examples. — Arts.  110-118.  Differentiation  along  an  axis.  Axes  of  a  Spherical 
Harmonic.  —  Art.  119.  Roots  of  a  Zonal  Harmonic.  Roots  of  a  Tesseral  Har- 
monic.    Nomenclature  justified. 

CHAPTER    VII. 
Cylindrical  Harmonics  (Bessel's  Functions) 219-237 

Art.  120.  Recapitulation.  Cylindrical  Harmonics  (Bessel's  Functions)  of  the 
zeroth  order;  of  the  uth  order;  of  the  second  kind.  General  solution  of  Bessel's 
Equation. — Art.  121.  Bessel's  Functions  as  definite  integrals.  Examples. — 
Art.  122.  Properties  of  Bessel's  Functions.  Semi-convergent  series  for  a  Bessel's 
Function.  Examples.  —  Art.  123.  Problem:  Stationary  temperatures  in  a 
cylinder  (o)  when  the  temperature  of  the  convex  stirface  is  zero;  {b)  when  the 
convex  surface  is  adiathermanous;  {(■)  when  the  convex  surface  is  exposed  to  air 
at  the  temperature  zero. — Art.  121.  Roots  of  Bessel's  functions.  —  Art.  125. 
The  integral  of  r  times  the  product  of  two  Cylindrical  Harmonics  of  the  zeroth 
order.  Example.  —  Art.  12().  Development  in  Cylindrical  Harmonic  Series. 
Formulas  for  the  coefficients.  Examples. — Art.  127.  Problem:  Stationary 
temperatures  in  a  cylindrical  shell.  Bessel's  Functions  of  the  second  kind 
employed.  Example:  Vibration  of  a  ring  membrane. — Art.  128.  Problem: 
Stationary  temperatures  in  a  cylinder  when  the  temperature  of  the  convex  surface 
varies  with  the  distance  from  the  base.  Bessel's  Functions  of  a  complex  variable. 
Examples.  —  Art.  129.  Problem:  Stationary  temperatures  in  a  cylinder  when 
the  temperatures  of  the  base  are  unsyuimetrical.  Bessel's  Functions  of  the  nth 
order  employed.  INIiscellaneotis  examples.  Bessel's  Functions  of  fractional 
order. 

CHAPTER    VIII. 

Laplace's  Equation  in  Curvilinear  Coordinates.      Ellipsoidal  Harmonics 238-2(30 

Arts.  130-131.  Orthogonal  Curvilinear  Coordinates  in  general.  Laplace's  Equa- 
tion expressed  in  terms  of  orthogonal  curvilinear  coordinates  by  the  aid  of  Green's 
theorem.  —  Arts.  132-135.  Spheroidal  Coordinates.  Laplace's  Equation  in 
spheroidal  coordinates,  in  normal  spheroidal  coordinates.  Examples.  Condition 
that  a  set  of  curvilinear  coordinates  should  be  normal.  Thermometric  Parameters. 
Particular  solutions  of  Laplace's  Equation  in  spheroidal  coordinates.  Spheroidal 
Harmonics.  Examples.  The  Potential  Function  due  to  the  attraction  of  an 
oblate  spheroid.     Solution  for  an  external   point.     Examples. —Arts.  130-141. 


TABLE    OF    CONTENTS.  ix 

PAGES 

Ellipsoidal  Coordinates.  Laplace's  Equation  in  ellipsoidal  coordinates.  Normal 
ellipsoidal  coordinates  expressed  as  Elliptic  Integrals.  Particular  solutions  of 
Laplace's  Equation.  Lame's  Equation.  Ellipsoidal  Harmonics  (Lame's  Func- 
tions). Tables  of  Ellipsoidal  Harmonics  of  the  degrees  1,  2,  and  3.  Lamp's 
Functions  of  the  second  kind.  Examples.  Development  in  Ellipsoidal  Harmonic 
series.  Value  of  the  Potential  Function  at  any  point  in  space  when  its  value  is 
given  at  all  points  on  the  surface  of  an  ellipsoid.  —  Akt.  142.  Conical  Coordinates. 
The  product  of  two  Ellipsoidal  Harmonics  a  Spherical  Harmonic.  —  Art.  143. 
Toroidal  Coordinates.  Laplace's  Equation  in  toroidal  coordinates.  Particular 
solutions.     Toroidal  Harmonics.     Potential  Function  for  an  anchor  rina;. 


CHAPTER    IX. 
Historical  Summary 207-275 


APPENDIX. 


Tables 


Table  I.  Surface  Zonal  Harmonics.  Argument  6. 
Table  II.  Surface  Zonal  Harmonics.  Argument  x. 
Table  III.    Hyperbolic  Functions 


Table  IV.  Roots  of  Bessel's  Functions 
Table  V.  Roots  of  Bessel's  Functions 
Table  VI.    Bessel's  Functions 


CHAPTER    I. 


INTRODUCTION. 


1.  In  many  important  problems  in  mathematical  physic;^  ,;we:  fire  ol>lig.exl, 
to  deal  with  'partial  differential  equations  of  a  comparatively  simple  form. 

For  example,  in  the  Analytical  Theory  of  Heat  we  have  for  the  change  of 
temperature  of  any  solid  dne  to  the  flow  of  heat  within  the  solid,  the  equation 

D,u  =  a\D'^u  +  D^u  +  D;n)*  [i] 

where  u  represents  the  temperatiire  at  any  point  of  the  solid  and  t  the  time. 

In  the  simplest  case,  that  of  a  slab  of  infinite  extent  with  parallel  plane 
faces,  where  the  temperature  can  be  regarded  as  a  function  of  one  coordinate, 
[i]  reduces  to 

D,n  =  a^I);u,  [ii] 

a  form  of  considerable  importance  in  the  consideration  of  the  problem  of  the 
cooling  of  the  earth's  crust. 

In  the  problem  of  the  permanent  state  of  temperatures  in  a  thin  rectangular 
plate,  the  equation  [i]  becomes 

D^u-{-  D-^u=^0.  [Ill] 

In  polar  or  spherical  coordinates  [i]  is  less  simple,  it  is 

A  n  =  '^.  [i>.0--^I>,. ")  +  ^^  A  (Sin  0  A  ^0  +  ^^  I>^  ^'.].  [IV] 

In  the  case  where  the  solid  in  question  is   a  sphere  and  the  temperature 

at  any  point  depends   merely  on  the   distance  of  the  point  from  the  centre 

fivl  reduces  to  n  /    \         o  -r,o,    \  ,-  -, 

•-     -'  Dt(ru)  =  a-I)^(ru)  .  [vj 

In  cylindrical  coordinates  [i]  becomes 

I)tn  =  a^  [D; n  +  -  I),,  v  +  -,  Z>|  n  +  Z*/  iq  .  [ vi] 

In  considering  the   flow   of   heat  in  a  cylinder   when  the  temperature  at 

any  point   depends    merely  on    the   distance  r  of   the    point    from   the    axis 

fvil  becomes  i 

I),u  =  a'\I)^-u-\--I),.u).  [vii] 

*  For  the  sake  of  brevity  we  shall  often  use  the  symbol  V^  for  the  operation  Da'-  +D,/-  +!),'-; 
and  with  this  notation  equation  [i]  would  be  written  DfU  =  a'^'V'^  u. 


2  INTRODUCTION.  [Art.   1. 

In  Acoustics  in  several  problems  Ave  have  the  equation 

Bly  =  (rD;i/;  [viii] 

for  instance,  in  considering  the  transverse  or  the  longitudinal  vibrations  of  a 
stretched  elastic  string,  or  the  transmission  of  plane  sound  waves  through 
,^,th,e  air.      ^  .  ^      .    . 

"I '' J'f  in,  e.dn§i^^{ng  the  transverse  vibrations  of   a  stretched  string  Ave  take 
.    ,  .accpJint  ,a£  .the  j-esistance  of  the  air  [a'iii]  is  replaced  by 

^'v:j-';>;'.^'::. .:.••'.  .•*      i)!>/ -^  2ki),i/  =  criJ^>/.  [ix] 

In  dealing  Avith  the  vibrations  of  a  stretched  elastic  membrane,  Ave  have  the 

equation 

^  ,        _         D^z  =  c^(I)^z  +  I)^z),  [X] 

or  in  cylindrical  coordinates 

Dfz  =  <^{p^z  +  J  D^^  + 1  Dlz).  [XI] 

In  the  theory  of  Potential  Ave  constantly  meet  Laplace's  Equation 

D-^  V  +  Dj  V  +  I)-;  V  =  0  [xii] 

or  V'V^O 

Avhich  in  spherical  coordinates  becomes 

7.  [>■!>; 0-r)  +  ^^  Z>, (sin 6 AT)  +  ^  I>,j;r]  -  0,  [xiii] 

and  in  cyUndrical  coordinates 

B::  V+ll)J'+  -,  I>lV-\-  D!  V  =  0.  [XIV] 

In  curvilinear  coordinates  it  is 

/'./-«.  [i'„(o;^,.r)+/',(^^„'-)  +  i>.,()r^i>.J-)]  =  o.     [xvi 

Avhere  /i  {x,y,z)  =  pi ,   /:,  {xjj.z)  =  p. ,  /;  (.r,//,^)  =  pg 

represent  a  set  of  surfaces  Avhich  cut  one  another  at  right  angles,  no  matter 
Avhat  values  are  given  to  pi,  pn,  and  pg;  and  Avhere 

h,'  =  (i),p,r  +  {i),^p,f  +  {iKp.y 
h^  =  {D,p,y  +  ipyp.:)^  +  (Ap2)^ 

ll^  =  (D^PsY  +  (^yPsY  +  (Aps)S 
and,  of  course,  must  be  expressed  in  terms  of  pi ,  p., ,  and  p. . 

If   it   happens   that    V-pi  =  0,    V-p..  =  0,    and   V^ps  —  0,    then  Laplace's 
Equation  [xv]  assumes  the  very  simple  form 


Chap.  L]  PARTICULAR   SOLUTIONS.  3 

2.  A  differential  equation  is  an  equation  containing  derivatives  or  differen- 
tials with  or  without  the  primitive  variables  from  which  they  are  derived. 

The  general  solution  of  a  differential  equation  is  the  equation  expressing  the 
most  general  relation  between  the  primitive  variables  which  is  consistent  with 
the  given  differential  equation  and  which  does  not  involve  differentials  or 
derivatives.  A  general  solution  will  always  contain  arbitrary  (i.  e.,  undeter- 
mined) constants  or  arbitrary  functions. 

A  particular  solution  of  a  differential  equation  is  a  relation  between  the 
primitive  variables  which  is  consistent  with  the  given  differential  equation, 
but  which  is  less  general  than  the  general  solution,  although  included  in  it. 

Theoretically,  every  particular  solution  can  be  obtained  from  the  general 
solution  by  substituting  in  the  general  solution  particular  values  for  the  arbi- 
trary constants  or  particular  functions  for  the  arbitrary  functions;  but  in 
practice  it  is  often  easy  to  obtain  particular  solutions  directly  from  the  differ- 
ential equation  when  it  would  be  difficult  or  impossible  to  obtain  the  general 
solution. 

3.  If  a  problem  requiring  for  its  solution  the  solving  of  a  differential  equa- 
tion is  determinate,  there  must  always  be  given  in  addition  to  the  differential 
equation  enough  outside  conditions  for  the  determination  of  all  the  arbitrary 
constants  or  arbitrary  functions  that  enter  into  the  general  solution  of  the 
equation;  and  in  dealing  with  such  a  problem,  if  the  differential  equation  can 
be  readily  solved  the  natural  method  of  procedure  is  to  obtain  its  general 
solution,  and  then  to  determine  the  constants  or  functions  by  the  aid  of  the 
given  conditions. 

It  often  happens,  however,  that  the  general  solution  of  the  differential  equa- 
tion in  question  cannot  be  obtained,  and  then,  since  the  problem  if  determinate 
will  be  solved  if  by  any  means  a  solution  of  the  equation  can  be  found  which 
will  also  satisfy  the  given  outside  conditions,  it  is  worth  while  to  try  to  get 
jjarticidar  solutions  and  so  to  combine  them  as  to  form  a  result  which  shall 
satisfy  the  given  conditions  without  ceasing  to  satisfy  the  differential  equation. 

4.  A  differential  equation  is  linear  when  it  would  be  of  the  first  degree  if 
the  dependent  variable  and  all  its  derivatives  were  regarded  as  algebraic 
unknown  quantities.  If  it  is  linear  and  contains  no  term  which  does  not 
involve  the  dependent  variable  or  one  of  its  derivatives,  it  is  said  to  be  linear 
and  homogeneous. 

All  the  differential  equations  collected  in  Art.  1  are  linear  and  homogeneous. 

5.  If  ci  value  of  the  dependent  variable  has  been  found  ivhich  satisfies  a 
given  homogeneous,  linear,  differential  equation,  the  product  formed  by  muliipdy- 
ing  this  value  by  any  constant  will  also  be  a  value ,  of  the  dependent  variable 
which  icill  satisfy  the  equation. 


4  INTRODUCTION.  [Art.  t>. 

For  if  all  the  terms  of  the  given  equation  are  transposed  to  the  first  mem- 
ber, the  substitution  of  the  lii-st-named  value  must  reduce  that  member  to 
zero;  substituting  the  second  value  is  equivalent  to  multiplying  each  term  of 
the  result  of  the  first  substitution  by  the  same  constant  factor,  which  there- 
fore may  be  taken  out  as  a  factor  of  the  whole  first  member.  The  remaining 
factor  being  zero,  the  product  is  zero  and  the  equation  is  satisfied. 

If  several  values  of  the  dependent  variable  have  been  found  each  of  tvhich 
satisfies  the  given  differential  equation^  their  sum  ivill  satisfy  the  equation  ;  for 
if  the  sum  of  the  values  in  question  is  substituted  in  the  equation  each  term 
of  the  sum  will  give  rise  to  a  set  of  terms  which  must  be  equal  to  zero,  and 
therefore  the  sum  of  these  sets  must  be  zero. 

6.  It  is  generally  possible  to  get  by  some  simple  device  ^yarticular  solutions 
of  such  differential  equations  as  those- we  have  collected  in  Art.  1.  The 
object  of  the  branch  of  mathematics  with  which  we  are  about  to  deal  is  to 
find  methods  of  so  combining  these  particular  solutions  as  to  satisfy  any  given 
conditions  which  are  consistent  with  the  nature  of  the  problem  in  question. 

This  often  requires  us  to  be  able  to  develop  any  given  function  of  the  varia- 
bles which  enter  into  the  expression  of  these  conditions  in  terms  of  normal 
forms  suited  to  the  problem  with  which  we  happen  to  be  dealing,  and  sug- 
gested by  the  form  of  particular  solution  that  we  are  able  to  obtain  for  the 
differential  equation. 

These  normal  forms  are  frequently  sines  and  cosines,  but  they  are  often 
much  more  complicated  functions  known  as  Legendre's  Coefficients,  or  Zonal 
Harmonics  ;  Laplace^s  Coefficients,  or  Spherical  Harmonics  ;  BesseVs  Funefions, 
or  Cylindrical  Harmonics  ;  Lame's  Functions,  or  Ellipsoidal  Harmonics,  &c. 

7.  As  an  illustration,  let  us  take  Fourier's  problem  of  the  permanent  state 
of  temperatures  in  a  thin  rectangular  plate  of  breadth  ir  and  of  infinite  length 
whose  faces  are  impervious  to  heat.  We  shall  suppose  that  the  two  long 
edges  of  the  plate  are  kept  at  the  constant  temperature  zero,  that  one  of  the 
short  edges,  which  we  shall  call  the  base  of  the  plate,  is  kept  at  the  tempera- 
ture unity,  and  that  the  temperatures  of  points  in  the  plate  decrease  indefi- 
nitely as  we  recede  from  the  base;  we  shall  attempt  to  find  the  temperature 
at  any  point  of  the  plate. 

Let  us  take  the  base  as  the  axis  of  X  and  one  end  of  the  base  as  the  origin. 

Then  to  solve  the  problem  we  are  to  find  the  temperature  u  of  any  point  from 

the  equation  r^o      i    7.0         n  r     -1   *  ^   i 

■■•  D-  u  -f  1)^1  It  =  0  [ill]  Art.  1 

subject  to  the  conditions       u  =  0  when  .r  =  0  (1) 

u  =  0           "  ,r  =  TT  ,  (2) 

u  =  0          "  y  =  00  (3) 

n  =  l          "  //  =  0.  (4) 


Chap.  I.]  KECTANGULAR    PLATE.  5 

We  shall  begin  by  getting  a  particular  solution  of  [m],  and  we  shall  use 
a  device  which  always  succeeds  when  the  equation  is  liiiear  and  homor/eneous 
and  has  constant  coefficients. 

Assume*  u  =  e°-y'^^^,  where  a  and  ft  are  constants,  substitute  in  [iii]  and 
divide  by  e'^y  +  ^-'\  and  we  have  a-  +  /3'  =  0.  If,  then,  this  condition  is  satis- 
fied ^l  =  e'^y  +  ^•''  is  a  solution. 

Hence  u  =  e'^^  -  "•''  t  is  a  solution  of  [m],  no  matter  what  value  may  be 
given  to  a. 

This  form  is  objectionable,  since  it  involves  an  imaginary.  We  can,  how- 
ever, readily  improve  it. 

Take  -u  ^  e'^y  e'^",  a  solution  of  [m],  and  zi  =  e'^^'e"'^',  another  solution 
of  [ill];  add  these  values  of  u  and  divide  the  sum  by  2  and  Ave  have 
e'^y  cos  ax.     (v.  Int.  Cal.  Art.  35,  [1].)     Therefore  by  Art.  5 

u  =  e'^y  cos  ax  (p) 

is  a  solution  of  [m].  Take  n  =  e'^y  e'^'  and  u  =  e°-y  e~'^\  subtract  the 
second  value  of  u  from  the  first  and  divide  by  21  and  we  have  e°-y  sin  ax. 
(v.  Int.  Cal.  Art.  35,  [2]).     Therefore  by  Art.  5 

u  =  e°-y  sin  ax  (Q) 

is  a  solution  of  [m]. 

Let  us  now  see  if  out  of  these  particular  solutions  we  can  build  up  a  solu- 
tion which  will  satisfy  the  conditions  (1),  (2),  (3),  and  (4). 

Consider  u  =  e°-y  sin  ax  .  (6) 

It  is  zero  when  x  =  0  for  all  values  of  a.  It  is  zero  when  a-  =  tt  if  a  is  a 
whole  number.  It  is  zero  when  3/  =  oo  if  a  is  negative.  If,  then,  we  write 
u  equal  to  a  sum  of  terms  of  the  form  Ae~"'''  sin  mx,  where  f/i  is  a  positive 
integer,  we  shall  have  a  solution  of  [iii]  which  satisfies  conditions  (1),  (2) 
and  (3).     Let  this  solution  be 

II  =  Aie~"  sin  x  +  A^e---'  sin  2x  +  Jge-""  sin  3x  -f- J^e"'*^  sin  4:X  -{-  -  •  ■   (7) 

Ai,  A2,  As,  Ai,  &c.,  being  undetermined  constants. 
When  1/  =  0  (7)  reduces  to 

II  —  Ai  sin  X  +  A„  sin  2x  -{-  A^  sin  ox  -\-  A^  sin  4.r  +  •  *  '  •  (8) 

If  now  it  is  possible  to  develop  unity  into  a  series  of  the  form  (8),  our 
problem  is  solved;  we  have  only  to  substitute  the  coefficients  of  that  series  for 

Ai,  A2,  A3,  &c.  in  (7). 

*  This  assumption  must  be  regarded  as  purely  tentative.     It  must  be  tested  by  substi- 
tuting in  the  equation,  and  is  justified  if  it  leads  to  a  solution, 
t  We  shall  regularly  use  the  symbol  i  f or  v^  —  1 . 


6  INTRODUCTION.  [Akt.  8. 

It  will  be  proved  later  that 

1  =  -|  sin  .r  +  o  sin  3x  +  ir  sin  5x  +  f  sin  T.r  +  •  •  •  ) 
for  all  values  of  x  between  0  and  tt;  hence  our  required  solution  is 
«  =  -     e~''  sin  .r  +  o"  e^^"  sin  3x -\-  y  e~'^''  sin  5x  +  =r  e~'^  sin  7x  -{-  ■  ■  •        (9) 

for  this  satisfies  the  differential  equation  and  all  the  given  conditions. 

If  the  given  temperature  of  the  base  of  the  plate  instead  of  being  unity 
is  a  function  of  x,  we  can  solve  the  problem  as  before  if  we  can  express  the 
given  function  of  cc .  as  a  sum  of  terms  of  the  form  A  sin  m  x,  where  m  is  a 
whole  number. 

The  problem  of  finding  the  value  of  the  2yote7itial  function  at  any  point  of 
a  long,  thin,  rectangular  conducting  sheet,  of  breadth  tt,  through  which  an 
electric  current  is  flowing,  when  the  two  long  edges  are  kept  at  potential  zero, 
and  one  short  edge  at  potential  unity,  is  mathematically  identical  with  the 
problem  we  have  just  solved. 

Example. 

Taking  the  temperature  of  the  base  of  the  plate  described  above  as  100° 
centigrade,  and  that  of  the  sides  of  the  plate  as  0°,  compute  the  temperatures 
of  the  points 

(-)(ia);   (^)(i,2);   (c)(|,3), 
correct  to  the  nearest  degree.  Ans.  (a)  25°;   (b)  15°;   (c)  6°. 

8.  As  another  illustration,  we  shall  take  the  problem  of  the  transverse 
vibrations  of  a  stretched  string  fastened  at  the  ends,  initially  distorted  into 
some  given  curve  and  then  allowed  to  swing. 

Let  the  length  of  the  string  be  /.  Take  the  position  of  equilibrium  of  the 
string  as  the  axis  of  X,  and  one  of  the  ends  as  the  origin,  and  suppose  the 
string  initially  distorted  into  a  curve  whose  equation  y  =f{x)  is  given. 

We  have  then  to  find  an  expression  for  ij  which  will  be  a  solution  of  the 
equation 

Bt^l/  =  a^D;y  [viii]  Art.  1, 

while  satisfying  the  conditions 

l,  =  0     when  .r  =  0  (1) 

,/^0         "  x^l  (2) 

v/=/(,r)    ''  f  =  0  (3) 

X»,y  =  0         "  t  =  0,  (4) 

the  last  condition  meaning  merely  that  the  string  starts  from  rest. 


Chap.  I.]  VIBRATING    STRING.  7 

As  in  the  last  problem  let  *  y  =^  e'^  +  ^*  and  substitute  in  [viii].     Divide 

by  e«^  +  ^«  and  we  have  yS^=  a^d^  as  the  condition  that  our  assumed  value  of 

y  shall  satisfy  the  equation.  _  ^^^at  ,r\ 

U  —  '^  (o; 

is,  then,  a  solution  of  (viii)  whatever  the  value  of  a. 

It  is  more  convenient  to  have  a  trigonometric  than  an  exponential  form  to 

deal  with,  and  we  can  readily  obtain  one.by  using  an  imaginary  value  for  a  in  (5). 

Replace  a  by  ai  and  (5)  becomes  ?/ =  Qi.^^o.t)ax^  ^  solution  of  [viii].      Replace 

a  by  —  ai  and  (5)  becomes  _?y  :^  e-C'^^nOa*^  another  solution  of  [viii].     Add 

these  values  of  y  and  divide  by  2  and  we  have  cos  a{x  ±  a£).       Subtract  the 

second  value  of  y  from  the  first  and  divide  by  2i  and  we  have  sin  a{x  ±  at). 

y  =  cos  a(.r  +  at) 
y  =  cos  a(.r  —  at) 
y  =  sin  a(x  -{-  at) 
y  =  sin  a(x  —  at) 

are,  then,  solutions  of  [vm].  Writing  y  successively  equal  to  half  the  sum 
of  the  first  pair  of  values,  half  their  difference,  half  the  sum  of  the  last 
pair  of  values,  and  half  their  difference,  we  get  the  very  convenient  particular 
solutions  of  [viii]. 

y  =  cos  ax  cos  aat 

y  =  sin  ax  sin  aat 

y  z=  sin  ax  cos  aat 

y  =  cos  ax  sin  aat . 
If  we  take  the  third  form 

y  =  sin  ax  cos  aat 

it  will  satisfy  conditions  (1)  and  (4),  no  matter  what  value  may  be  given  to 

a,  and  it  will  satisfy  (2)  if  a  :=  -r-  where  m  is  an  integer. 

If  then  we  take 

.     .     TTX        irat   ,     .     .     2'Trx        lirat  ,     ,     .     Zirx        Sirat   , 
y:=A-i^  sm  —  cos  — — \-  Ao  sm  —j—  cos  — 1-  A3  sm  — r—  cos  — 1 (6) 

where  Aj ,  Ao ,  A3  •  •  •  are  undetermined  constants,  we  shall  have  a  solution  of 
[viii]  which  satisfies  (1),  (2),  and  (4).     When  t  =  0  it  reduces  to 

y  =  Ai  sm  —-{-  A2  sm  -—  +  As  sm  -y-  +  •  •  •  (7) 

If  now  it  is  possible  to  develop  f(x)  into  a  series  of  the  form  (7),  we  can 
solve  our  problem  completely.  We  have  only  to  take  the  coefficients  of  this 
series  as  values  of  Ai,  A2,  A3  .  .  .  in  (6),  and  we  shall  have  a  solution  of 
[viii]  which  satisfies  all  our  given  conditions. 

*  See  note  on  page  5. 


8  INTRODUCTION.  [Art.  9. 

In  each  of  the  preceding  problems  the  normal  function,  in  terms  of  which  a 
given  function  has  to  be  expressed,  is  the  sine  of  a  simple  multiple  of  the 
variable.  It  would  be  easy  to  modify  the  problem  so  that  the  normal  form 
should  be  a  cosine. 

-We  shall  now  take  a  couple  of  problems  which  are  much  more  complicated 
and  where  the  normal  function  is  an  unfamiliar  one. 

9.  Let  it  be  required  to  find  the  potential  function  due  to  a  circular  wire 
ring  of  small  cross  section  and  of  given  radius  c,  supposing  the  matter  of  the 
ring  to  attract  according  to  the  law  of  nature. 

We  can  readily  find,  by  direct  integration,  the  value  of  the  potential  function 
at  any  point  of  the  axis  of  the  ring.     We  get  for  it 

where  M  is  the  mass  of  the  ring,  and  x  the  distance  of  the  point  from  the 
centre  of  the  ring. 

Let  us  use  spherical  coordinates,  taking  the  centre  of  the  ring  as  origin  and 
the  axis  of  the  ring  as  the  polar  axis. 

To  obtain  the  value  of  the  potential  function  at  any  point  in  space,  we  must 
satisfy  the  equation 

vD';  (r  V)  +  J^^  Deism  OBe  V)  +  ^^  Z>,^  V=  0,         [xiii]  Art.  1, 
subject  to  the  condition 

V=j;^^^,     ^vhen     ^  =  0.  (1) 

From  the  symmetry  of  the  ring,  it  is  clear  that  the  value  of  the  potential 
function  must  be  independent  of  ^,  so  that  [xiii]  Avill  reduce  to 

rl)::  (r  V)  +  ^^^  A  (sin  6  A  V)  =  0  .  (2) 

We  must  now  try  to  get  particular  solutions  of  (2),  and  as  the  coefficients 
are  not  constant,  we  are  driven  to  a  new  device. 

Let  *  V:=  r'"  P,  Avhere  P  is  a  function  of  6  only,  and  m  is  a  positive  integer, 
and  substitute  in  (2),  which  becomes 


.  (m  +  1);-'"P  +  ^— „  De  (sin  $DgP)=0. 
*  See  note  on  page  5  . 


Chap.   I.]  POTENTIAL    DUE    TO    WIRE    KING.  9 

Divide  by  r'"  and  use  the  notation  of  ordinary  derivatives  since  P  depends 
upon  6  only,  and  we  have  the  equation 

-(-  +  1)  P  +  ^^   -^-^^ =  0  ,  (3) 

from  which  to  obtain  P. 

Equation  (3)  can  be  simplified  by  changing  the  independent  variable.     Let 
a-  =  cos  0  and  (3)  becomes 

I  [  <i- ■'■■^>f  ]  +  "'("' +  i'^="-  w 

Assume  *.  now  that  F  can  be  expressed  as  a  sum  or  as  a  series  of  terms 
involving  whole  powers  of  x  multiplied  by  constant  coefficients. 
Let   P  =  2  ((„  a;"   and  substitute  this  value  of  P  in  (4).     We  get 

2  [h  (n  —  1)  ((„  X"  -'^—11  {)i  +  1)  a„  X"  +  m,  {in  +  1)  a„  cc"]  =  0  ,  (5) 

where  the  symbol  2  indicates  that  we  are  to  form  all  the  terms  we  can  by 
taking  successive  whole  numbers  for  n. 

As  (5)  must  be  true  no  matter  what  the  value  of  .r,  the  coefficient  of  any 
given  power  of  x,  as  for  instance  x'',  must  vanish.     Hence 

(A'  +  2)  (/.■  +  1)  a,  ^ ,  -  /.■  (7.-  +  1 )  a,  +  >n  (m  +  1)  a,  =  0  (6) 

and  a^+o  = (k-\-l)  (k  +2) "'^-  *  (^^ 

If  now  any  set  of  coefficients  satisfying  the  relation  (7)  be  taken,  P  =  S  '(k^^ 
will  be  a  solution  of  (4). 

If  l-^m,     r,,^,  =  0,     ..,,,=:  0,     &c. 

Since  it  will  answer  our  purpose  if  we  pick  out  the  simplest  set  of  coefficients 
that  will  obey  the  condition  (7),  we  can  take  a  set  including  a„^. 
Let  us  rewrite  (7)  in  the  form 

(/.■  +  2)(/.'+l)  ^_, 

"'  ~        (m  -  k)  (m  +  A-  +  1)  ''*■  +  2  •  (»; 

We  get  from  (8),  beginning  with    k  =  m  —  2, 

ID  (ill   1) 

""'--  ^  ~  2.  (2///  —  1)  ■'"' 

_  III  (ill  —  1)  (ill  —  2)  (m  —  3) 
«™_4  —       2A.(2iii  —i)(2iii  —  3)      "'« 

_        m  (m  —  1)  (H^  —  2)  (in  —  3)  (m  —  4)  (m  —  5) 
«^ _ e  —  2. 4. 6.  (21)1  -  1)  {2vi  —  3)  (2m  —  5)         ^*'»  '    *^^' 

*  See  note  on  page  5. 


10  INTRODUCTION.  [Art.  9. 

If  m  is  even  we  see  that  the  set  will  end  with  ^o ,  if  ^  is  odd,  with  «! . 

r  VI  (m  —  1)  „   ,    m  (m.  —  1)  (m  —  2)  (m  —  3)  ~| 

r  —  a^  ^x         2.(2;»,  -  1)  •''        ^       2.4.(2w  -  1)  (2m  -  ^)      ''  J 

where  a„^  is  entirely  arbitrary,  is,  then,  a  solution  of  (4).     It  is  found  con- 
venient to  take  «„j  equal  to 

{2m  —  1)  (2vi  -  3)  •  •  •  1 
m\ 
and  it  can  be  shown  that  with  this  value  of  <•/,„  P ^  1  when  x  ^=\. 

P  is  a  function  of  x  and  contains  no  higher  powers  of  x  than  a-'".  It  is 
usual  to  write  it  as  P„,  {x). 

We  proceed  to  compute  a  few  values  of  P„,  (x)  from  the  formula 

(2m  —  1)  (2m^  —  3)  •  •  •  1  r  m  Cm.  —  1) 

^^  '  ml  L  2. (27)1  —  1) 

m(m-l)(m-2)(m-3)  _       n 
"^     2. 4.  (2m  -  1)  (2m  —  3)      ^  J  •  C^^ 

We  have: 

Po(:r)  =  1  or  Po(cos  6)  =  1 

p^(x)  ^x  "  Pi  (cos  0)  =  GOSO 

p^(x)  =  i  (3x2  _  1)  a  p^(cos  e)  =  i(3  cos2  ^  -  1) 

P^(x)  =  i  (5a;3  —  3.x)        "  P3(cos  6)  =  i  (5  cos^^  —  3  cos  6) 

p}J)  =  X  (35^4  -  30^2  +  3)  or 

p^(cos  0)  =  i(35  cos"^  —  30  cos'^^  +  3) 
P^(x)  =  i  (63^5  —  70^3  +  15.X-)  or 

P,(cose)  =  i(63  cos^^  —  70  cos^^  +  15  cos^) . 

We  have  obtained  F  =  P,^(x)  as  a  particular  solution  of  (4)  and 
P  =  P^  (cos  6)  as  a  particular  solution  of  (3).  P,„  (.r)  or  P,„  (cos  6)  is  a 
new  function,  known  as  a  Lef/endre's  Coefficient,  or  as  a  Surface  Zonal  Har- 
monic, and  -occurs  as  a  normal  form  in  many  important  problems. 

V  =  r'^'P^^  (cos  $)  is  a  particular  solution  of  (2)  and  r'»P,„  (cos  0)  is  some- 
times called  a  Solid  Zonal  Harmonic. 

We  can  now  proceed  to  the  solution  of  our  original  problem. 

r=  AroPo(cos  ^)  +  .liV-Pi(cos  ^)  +  J2»-'A(cos  6)  +A^r^P^(cos  6) -\ (11) 

where  Aq,  A^,  A^,  &c.,  are  entirely  arbitrary,  is  a  solution  of  (2)  (v.  Art.  5). 
When  ^  =  0  (11)  reduces  to 

r=A^,  +  A,r+A,r'-\-A,r'+---, 

since,  as  we  have  said,  P„,  (x)  =  1  when  x  =  1,  or  P„j  (cos  $)  =  1  when  ^  =:  0. 
By  our  condition  (1)  J^^ 

when  ^  =  0.  I    -t-'  ) 


(10) 


Chap.  I.]  ZONAL    HARMONICS.  11 

By  the  Binomial  Theorem 

provided  >•  <  <•.     Hence 

is  our  required  solution  if  y  <  r ;  for  it  is  a  solution  of  equation  (2)  and  satis- 
fies condition  (1). 

Example. 
Taking  the  mass  of  the  ring  as  one  pound  and  the  radius  of  the  ring  as  one 
foot,  compute  to  two  decimal  places  the  value  of  the  potential  function  due  to 
the  ring  at  the  points 

(,,)    (,.  =  .2,^  =  0);  (d)   (r  =  .Q,0  =  O);  (/)  ^,  =  .6,  ^  =  0; 

(h)   (.  =  .2,^  =  ^);  (.)    (.  =  .r,^  =  ~);  (r/)  (r  =  .6,0^1); 

(r)   (r  =  2,  ^  =  ");  ^^»'-  ('0  "^S;  ('')  -99;  (r)  1.01;  (rf)  .86; 

V  -/  (,)  .90;  (/)  1.00;    (,y)  1.10. 

The  unit  used  is  the  potential  due  to  a  pound  of  mass  concentrated  at  a  point 
and  attracting  a  second  pound  of  mass  concentrated  at  a  point,  the  two  points 
being  a  foot  apart. 

10.  A  slightly  different  problem  calling  for, development  in  terms  of  Zonal 
Harmonics  is  the  following: 

Required  the  permanent  temperatures  within  a  solid  sphere  of  radius  1, 
one  half  of  the  surface  being  kept  at  the  constant  temperature  zero,  and  the 
other  half  at  the  constant  temperature  unity. 

Let  us  take  the  diameter  perpendicular  to  the  plane  separating  the  unequally 
heated  surfaces  as  our  axis  and  let  us  use  spherical  coordinates.  As  in  the 
last  problem,  we  must  solve  the  equation 

''^'^''"^  +  ^mO  ^' ('^'' ^  ^' "^  +  ^^hTO  ^*  "  ^  ^-  t""'"^  ^''^-  ^ 

which  as  before  reduces  to 

rD;\ru)  +  ^^  A(sin  6  A  ^0  =  0  (1) 

from  the  consideration  that  the  temperatures  must  be  independent  of  <^. 
Our  equation  of  condition  is 

It  =  1  from  ^  =  0  to  0  =  ^  and  u  =  0  from  ^  =  J  to  ^  =  tt  ,        (2) 
when   /■  ^  1. 


12  INTRODUCTION.  [Art.  10. 

As  we  have  seen  u  =  r"'P,„(cos  0)  is  a  particular  solution  of  (1),  m  being 
any  positive  whole  number,  and 

i(  =  Ao  roPo  (cos  0)+A,  rPi  (cos  $)  +  A.  r'Fo^  (cos  0)  +  A^  y^P^  (cos  6) -\ (3) 

where  Aq,  A-^,  A,  A^  ■  ■  •  are  undetermined  constants,  is  a  solution  of  (1). 
When  r  ^1  (3)  reduces  to 

u  =^  AoPo  (cos  6)  +  ^1  Pi  (cos  0)  +  APo  (cos  0)  +  ^.Pg  (cos  6)  H (4) 

If  then  we  can  develop  our  function  of  6  which  enters  into  equation  (2)  in 
a  series  of  the  form  (4),  we  have  only  to  take  the  coefficients  of  that  series 
as  the  values  of  Aq,  A^,  A^,  &c.,  in  (3)  and  we  shall  have  our  required  solution. 

11.  As  a  last  example  we  shall  take  the  problem  of  the  vibration  of  a  stretched 
circular  membrane  fastened  at  the  circumference,  that  is,  of  an  ordinary  drum- 
head. We  shall  suppose  the  membrane  initially  distorted  into  any  given  form 
which  has  circular  symmetry  *  about  an  axis  through  the  centre  perpendicular 
to  the  plane  of  the  boundary,  and  then  allowed  to  vibrate. 

Here  we  have  to  solve 

PA.  =  .^  (P,r'^  +  I  D,.z  +  \,  D^^  [XI]  Art,  1 

subject  to  the  conditions 

z  =/(!■)  when  ^  =  0  (1; 

D,z  =  0  ''  /  =  0  '                       (2) 

^  =  0  -  .  =  ...  (3) 

From  the  symmetry  of  the  supposed  initial  distortion  z  must  be  independ- 
ent of  ^,  therefore  [xi]  reduces  to 

P,2,-  =  r2  (n^rz  +  -  D,.z\  (4) 

and  this  is  the  equation  for  which  we  wish  to  find  a  particular  solution. 

We  shall  employ  a  device  not  unlike  that  used  in  Art.  9. 

Assume  f  z  =^  R.T  where  P  is  a  function  of  r  alone  and  T  is  a  function  of 
t  alone.     Substitute  this  value  of  z  in  (4)  and  we  get 


bd;'t  =  c^T  (D,m  +  -d^r\ 


1    (PT  _  1  uPB       1  dR 
'^T'df'  ~E  \d7  +  r  7/7 ; 


The   second  member  of  (5)  does  not  involve  t,  therefore  its  equal  the  first 
member  must  be  independent  of  f.     The  first  member  of  (5)  does  not  involve 

*  A  function  of  the  coordinates  of  a  point  has  circular  symmetry  about  an  axis  when  its 
value  is  not  affected  by  rotating  the  point  through  any  angle  about  the  axis.  A  surface  has 
circular  symmetry  about  an  axis  when  it  is  a  surface  of  revolution  about  the  axis. 

t  See  note  on  page  5. 


Chap.   I.]  VIBRATING   DRUMHEAD.  13 

r,  and  consequently  since  it  contains  neither  t  nor  r,  it  must  be  constant.      Let 
it  equal  —  /ot^  where  fx  of  course  is  an  undetermined  constant. 
Tlien  (5)  breaks  up  into  the  two  differential  equations 

-,  +  ^^<'T=i)  (6) 

d'^R.        1  (IE 

^^+-^  +  /.-^A>  =  ().  (7) 

(6)  can  be  solved  by  familiar  methods,  and  we  get  T  =  cos  ixct  and  T  =  sin  /xd 
as  simple  particular  solutions  (v.  Int.  Cal.  p.  319,  §  21). 

To  solve  (7)  is  not  so  easy.     We  shall  first  simplify  it  by  a  change  of  inde- 
pendent variable.     Let   r  =  -  ■   (7)  becomes 

cPB       1  dR 

(kr^         X  dx     '  *  ^ 

Assume,  as  in  Art.  9,  that  R  can  be  expressed  in  terms  of  whole  powers  of 
X.     Let  R  =  ^a,^x"   and  substitute  in  (8).     We  get 

2  [«('«  —  l)a„x^'~^  -\-  7ia„x"~'^  -j-  aj,x"^  =  0  , 

an  equation  which  must  be  true  no  matter  what  the  value  of  x.     The  coeffi- 
cient of  any  given  power  of  x,  as  .t*'-^,  must,  then,  vanish,  and 

k(k  —  l)a^.  -f  ka^.  +  r?^._,  =  0 
or  A--(^^.  -f  (f,_,  =  0 

whence  we  obtain  %._2  =  —  k'^a^.  (9) 

as  the  only  relation  that  need  be   satisfied  by  the  coefficients   in  order  that 
i?  =  2  a^.x'^  shall  be  a  solution  of  (8). 

If  A-  =  0,     a^._,_=0,     a,._,=  0.     &c. 

We  can  then  begin  with  ^  ^  0  as  our  lowest  subscript. 

From  (9)  a,  =  -'^- 

Then  a^  =  —  -^^ 


9'-' 


.4^.6^ 


&c. 


Hence  R  =  a^  fl  -  |o  +  gfj.  -  22.42,(3-2  +     '  '  1 

where  «o  may  be  taken  at  pleasure,  is  a  solution  of  (8),  provided  the  series  is 
convergent. 


14  IXTEODUCTION.  [Art.  11. 

Take  ao=^l,  and  then   li  =  Jq(x)   where 

Jq  (a')  =  1  —  22  +  2^  ~  2-.4-.6-  "^  2-. 4-. 6^. 8-  ~  '  '  *  (10) 

is  a  solution  of  (8). 

t/o  (^)  is  easily  shown  to  be  convergent  for  all  values  real  or  imaginary  of  x. 
since  the  series  made  up  of  the  moduli  of  the  terms  of  Jo  (a;)  (v.  Int.  Cal. 
Art.  30) 

1  ~r  2^  "T  2^^    I   2-  -4"-  6^    i    '  '  ' ' 

where  r  is  the  modulus  of  x,  is  convergent  for  all  values  of  r.     For  the  ratio 

of  the  n  +  1  st  term  of  this  series  to  the  7i  th  term  is  and  approaches 

4«- 
zero  as  its  limit  as  n  is  indefinitely  increased,  no  matter  what  the  value  of  ;■. 
Therefore  Jq  (x)  is  absolutely  convergent. 

Jq  (x)  is  a  new  and  important  form.  It  is  called  a  BesseVs  Function  of  the 
zero  th  order,  or  a  Cylindrical  Harmon ic. 

Equation  (8)  was  obtained  from  (7)  by  the  substitution  of   x  =  fir,    therefore 

J?    r/  ^    1    (/^'■■)'    (M^     (f'^-y  , 

B  =  J,(ixr)  =  1  -  -2^  +  2U"^  -  -22;42;go  -f  •  •  • 

is  a  solution  of  (7),  no  matter  what  the  value  of  /*,  and  z  '=  Jq  {fir)  cos  fict 
or  z  =  Jo(f^r)  sin  fict  is  a  solution  of  (4). 

~  ^  Jo  (/^'")  cos  fict  satisfies  condition  (2)  whatever  the  value  of  /x.  In 
order  that  it  should  also  satisfy  condition  (3)  fi  must  be  so  taken  that 

Jo(A^^0  =  O;  (11) 

that  is,  fi  must  be  a  root  of  (11)  regarded  as  an  equation  in  fx. 

It  can  be  shown  that   Jq  (x)  =  0   has  an  infinite  number  of  real  jjositive 
roots,  any  one  of  which  can  be  obtained  to  any  required  degree  of  approxima- 
tion without  serious  difficulty.     Let  Xy,  X2,  a-g,  •  •  •  be  these  roots.     Then  if 
,-(■1  .r,  a-3 

z  =  Ai Jo  (/^i  r)  cos  /^i  rf  +  A.yJo  (fjL2  r)  cos  fi.  ct  +  A^ Jo  (^3  »■)  cos  /Ug  c^  +  •  • ' »   (12) 

where  A^,  A^,  A^,  &c.,  are  any  constants,  is  a  solution  of  (4)  which  satisfies 
conditions  (2)  and  (3). 

When  t  =  0  (12)  reduces  to 

z  =  A,  Jo  (fi,  r)  +  A,  Jo  (fi,  r)  +  A,  J,  (f^s  r)  +  •  •  •  .  (13) 

If  then  /(?•)  can  be  expressed  as  a  series  of  the  form  just  given,  the  solution 
of  our  problem  can  be  obtained  by  substituting  the  coefficients  of  that  series 
for  .li,  A^,  As.  &c.,  in  (12). 


Chap.  I.]  DISCUSSION   OF   METHODS.  15 

Example. 

The  temperature  of  a  long  cylinder  is  at  first  unity  throughout.  The  convex 
surface  is  then  kept  at  the  constant  temperature  zero.  Show  that  the  tem- 
perature of  any  point  in  the  cylinder  at  the  expiration  of  the  time  t  is 

where  /Xi,  fx^,  &c.,  are  the  roots  of  Jii{fJic)  =  0,  and  where 

1  =^A,J,{fx,r)  +  A,J,{iJ.,r)  +  A,J,{fjL,r)  +  •  •  •, 
c  being  the  radius  of  the  cylinder. 

12.  Each  of  the  five  problems  which  we  have  taken  up  forces  upon  us  the 
consideration  of  the  development  of  a  given  function  in  terms  of  some  normal 
form,  and  in  two  of  them  the  normal  form  suggested  is  an  unfamiliar  function. 
It  is  clear,  then,  that  a  complete  treatment  of  our  subject  will  require  the  inves- 
tigation of  the  properties  and  relations  of  certain  uqw  and  important  functions, 
as  well  as  the  consideration  of  methods  of  developing  in  terms  of  them. 

13.  In  each  of  the  problems  just  taken  up  we  have  to  deal  with  a  homo- 
geneous linear  partial  differential  equation  involving  two  independent  vari- 
ables, and  we  are  content  if  we  can  obtain  particular  solutions.  In  each  case 
the  assumption  made  in  the  last  problem,  that  there  exists  a  solution  of  the 
equation  in  which  the  dependent  variable  is  the  product  of  two  factors  each  of 
which  involves  but  one  of  the  independent  variables,  will  reduce  the  question 
to  solving  two  ordinary  differential  equations  which  can  be  treated  separately. 

If  these  equations  are  familiar  ones  their  solutions  can  be  written  down  at 
once;  if  unfamiliar,  the  device  iised  in  problems  3  and  5  is  often  serviceable, 
namely,  that  of  assuming  that  the  dependent  variable  can  be  expressed  as  a 
sum  or  series  of  terms  involving  whole  powers  of  the  independent  variable, 
and  then  determining  the  coefficients. 

Let  us  consider  again  the  equations  used  in  the  first,  second  and  third 
problems. 

{a)  D^u  -f  D^u  =  0  (1) 

Assume    u  =  X.  Y  where  X  involves  x  but  not  ij,  and  Y  involves  //  but  not  x. 
Substitute  m  (1) ,  FD^^  x  -|-  A'X»/ 1'  =  0, 

or,  since  we  are  now  dealing  with  functions  of  a  single  variable, 
1  d'X  .    1  d-'Y 


(2) 


Xd^^-^ 

Ydf'-''' 

1  d'Y 

1  d'X 

Y  df  - 

X  dx^  ■ 

16  INTRODUCTION.  [Akt.  13. 

Since  the  first  member  of  (2)  does  not  contain  a?,  and  the  second  member 
does  not  contain  y,  and  the  two  members  must  be  identically  equal,  neither  of 
them  can  contain  either  x  or  y,  and  each  must  be  equal  to  a  constant,  say  d\ 

Then  ^^^Y=0  (3) 

and  ^T  +  a-U'=0;  (4) 

and  if  (3)  and  (4)  can  be  solved,  Ave  can  solve  (1).     They  have  for  their  com- 
plete solutions  p  ^  ^^^y  -\-  Be-'^y 

and  A'  =  C  sin  a:r  +  I)  cos  ax  .        (v.  Int.  Cal.  p.  319,  §  21.) 

Hence    F=  e"^'  and   F=  e~"^   are  particular  solutions  of    (3),    A^=  sin  ace 
and  A'=  cos  ax  are  particular  solutions  of  (4),  and  consequently 

u  =  f'°-v  sin  ax  ,     ti  ^^  e'^^  cos  ax  ,      u  =  e~°-y  sin  ax  ,  and  ?f  ^=  e~'^y  cos  ax 

are  particular  solutions  of  (4).     These  agree  with  the  results  of  Art.  7. 

(l>)  D-^,1  =  a'Dly  (1) 

Assume  //  =  T.X  where  T  is  a  function  of  t  only  and  X  a  function  of  x 
only;  substitute  in  (1)  and  divide  by  (i^TX.     We  get 

1    d'^T  _  1  d}X 
a^  'df'  ~X~di^  '  (^) 

1  d^X 

hence  as  in  the  last  case    -y-f^  is  a   constant;    call  it  —  a^,  and  (2)  breaks 

up  into  d'^X    ,      , ,,       ^ 

^  ^^  +  a^A'=0  (3) 

d'^T 

_^  +  «V^=0.  (4) 

The  complete  solutions  of  (3)  and  (4)  are 

X^  A  sin  ax  +  B  cos  ax 

and  T  =  C  sin  ar/?-  +  D  cos  aai,    (v.  Int.  Cal.  p.  319,  §  21). 

2/  =  sin  aa'cos  ao^    _?/ ^  sin  a^c  sin  ao^?,    ?/ =:  cos  a.r  cos  art/,    // =  cos  aa;  sin  aa# 

are  particular  solutions  of  (1),  and  agree  with  the  results  of  Art.  8. 

(c)  rD:-{r  V)  +  -^q  A  (sin  6  DeV)  =  0  .  (1) 

Assume  V=  R.®  where  B  involves  r  alone,  and  ©  involves  6  alone;  sub- 
stitute in  (1),  divide  by  B.®,  and  transpose;  we  get 

/  d®\ 
r  cP(rB)  _  _  ^_  'T^'^W 
B     dr'^      ~       0sin^  dO         '  (^) 


Chap.  I.]  SOLUTION   AS    A   PRODUCT.  17 

Since  by  the  reasoning  used  in  («)  and  (b)  each  member  of  (2)  must  be  a  con- 
stant, say  a^,  we  have 

d-'(rE) 
'     d^^  =  «'^  (3) 

and  1      H''''^d0) 


sin^  dO  +«^®  =  0-  (4) 

(3)  can  be  expanded  into 

dm  dB 

,._  +  2r^-a-^A'  =  0.  (5) 

(5)  can  be  solved  (v.  Int.  Cab  p.  321,  §  23),  and  has  for  its  complete  solution 
B  =  Ar'"  +  B?-"  , 

where  m  ^  —  -J-  +  v'  a'^  +  i  and         ?i  =  —  ^  —  V  «"^  +  i  • 

Hence    n  =  —  m  —  1,    and    a^   may  be  written    ))i(^m,  -\-  1),   m   being  wholly 
arbitrary;  and 

B  =  Ar'"  +  Br-'"-^  . 

B^V",        and        B^-f^-^ 

are,  then,  particular  solutions  of 

dm  dB 

r'  -^  +  2r  -^-  -  m(7,i  +1)B  =  0.  (6) 

With  the  new  value  of  a^  (4)  becomes 
/  d®\ 

SrT  ^-irO +  ^K-  +  1)0  =  0  .  (7) 

which  has  been  treated  in  Art.  9  for  the  case  where  m  is  a  positive  integer, 
and  the  particular  solution   0  =  P,„  (cos  0)    has  been  obtained. 

Hence  V  =  r'"P„^  (cos  0) 

and  r=-^-,P,„(Gose), 

m  being  a  positive  integer,  are  particular  solutions  of  (1).     The  first  of  these 
was  obtained  in  Art.  9,  but  the  second  is  new  and  exceedingly  important. 

14.  The  method  of  obtaining  a  particular  solution  of  an  ordinary  linear 
differential  equation,  which  we  have  used  in  Articles  9  and  11,  is  of  very 
extensive  application,  and  often  leads  to  the  general  solution  of  the  equation 
in  question. 


18  INTKODUCTiO^'.  [Akt.  U 

As  a  very  simple  example,  let  us  take  the  equation  Art.  13  (a)  (4),  which 
we  shall  write 

£;  +  "-  =  ".  (i> 

Assume  that  there  is  a  solution  Avhich  can  be  expressed  in  terms  of  powers 
of  x;  that  is,  let  .i  =  2«„-v",  Avhere  the  coefficients  are  to  be  determined. 
Substitute  this  lvalue  for  z  in  (1)  and  we  get 

2  [n(i/  —  T)a„x"--  +  a-a„x"']  =  0  . 

Since  this  equation  must  be  true  from  its  form,  without  reference  to  the  value 
of  X,  that  is,  since  it  must  be  an  identical  equation,  the  coefficient  of  each 
power  of  X  must  equal  zero,  and  we  have 

(»  +  !)(»  +  2) 
whence  "„  = o «„4.2 


is  the  only  relation  that  need   hold   between  the  coefficients   in   order   that 
2:  =  2  a^x"   should  be  a  solution  of  (1). 

If  «  +  2  =  0    or    «  +  1  =  0  ,  "„  will  be  zero  and  «„_..,  rt„_4,  &c.,  will  be 
zero.     In  the  first  case  the  series  will  begin  with  Uq,  in  the  second  with  Oj . 


If  we  begin  with  a^  we  have 


and 


a* 

4-,  «o  ,                      fte  = 

a" 

-  ^;  «o  .. 

,  &c.,  . 

:  +   4:        6!  + 

...) 

(2> 

—  «o  COS  ax 

(3> 

is  a  particular  solution  of  (1). 
If  we  begin  witii  «!  we  have 


a'  a" 

«7    =    =^.     rtl 


,!"l 


&C.,    .  .  . 

(a-x"        wx"        a"!-  \ 


Chap.  I.]  SOLUTION    IN    POWER    SERIES.  19 

is  a  solution  of  (1) ;  ai  can  be  taken  at  pleasure.     Let  «i  =  a,  (4)  becomes 

aV       a^x^       a^.r'' 

,  =  a.r  -    .-  +  ^-j-  _  -^  +  .  .  . 

or  z  =  sin  ax 

which,  then,  is  a  particular  solution  of  (1). 

z  =  A  sin  ax  +  B  cos  ax  (5) 

is,  then,  a  solution  of  (1),  and  since  it  contains  two  arbitrary  constants  it  is 
the  general  solution. 

15.  As  another  example  we  will  take  the  equation 

cPz  dz 

^'  ,7-2  +  ^-^^-  >"("'  +  1)^  =  0  ,  (1) 

which  is  in  effect  equation  (6),  Art.  13  (c),  and  let  m  be  a  positive  integer. 
Assume   s  ==  2  «„;>■"   and  substitute  in  (1).     We  get 

2  [n(;>i  +  1)  —  vi(m  +  1)]  a„x"  =  0  . 

This  is  an  identical  equation,  therefore 

[«(«  +  !)-  w(;^^  +  l)]r.„  =  0. 

Hence  a„  =  0  for  all  values  of  u  except  those  which  make 

n(n  +  1)  —  m(m.  +  1)  =  0  , 

that  is,  for  all  values  of  n  except  n  =:  »>,    and    n-  =  —  m  —  1 .     Then 

z=Ax'"  -{-  Bx-'"-'^  (2) 

is  the  general  solution  of  (1)  and 

z  =  X'"     and     z  =  --^-, 

are  particular  solutions.     If  m,  is  not  a  positive  integer  this  method  will  not 
lead  to  a  result,  and  we  are  driven  back  to  that  employed  in  Art.  13  (c). 

16.  Let  us  now  take  the  equation 

Txl(^--"^dx]+^''(^"+^^'  =  ^  (1) 

which  is  in  effect  equation  (4),  Art.  9,  and  is  known  as  Legendre's  Equation. 
(1)  may  be  written 

(^  -  ^')  £  ~  -^  S + "'('"' + ^^'^ = ^  •  (2) 


20  INTRODUCTION.  [Art.  16. 

Assume    .-:  :=  %  a^x"    and  substitute  in  (2).     We  get 

2  {n^^a  —  l)a„  .r"-2  +  [w(»;  +  1)  —  «("  +  l)]^'„a'"}  =  0  . 
Hence  {n  +  1)  (m  +  2)a„  ^ ,  +  [_m{m  +  1)  -  «(m  +  1)] «„  =  0  , 

(.+!)(» +2)  _  .^3^ 


"^  "«  -        /»(»^  +  l)- ?;(/; +1)  ""  +  2- 

If  o^^  =  0,  then  (/„  _  2  =  0,  a,^  _  ^  =  0,  &c. ;  but  ^^,  =  0  if  n  =  —  2  or  n  =  —  l. 

For  the  first  case  we  have  the  sequence  of  coefficients 

m{in  —  2)  {m  +  1)  (m  +  3) 
fH  = jj  ^^0 

m(m  -  2)  { //^  -  4)  {111  +  1)  (»^  +  3)  {m  +  5) 

Let  us  take  (Iq,  which  is  arbitrary,  as  1.     Then    z  =  2^m  (-0  '^vhere 

r-          m(m+l)    .,       w(//^  -  2)  (/>/ + 1)  ("^  +  3)    ^  ~|        /ix 

i>;« (^0  =  |_1 2! '"  +  ~  II  .Y  -  •  •  •  J        (4) 

is  a  solution  of  Legenclre's  Equation  if  ^j„,  (a-)  is  a  finite  sum  or  a  convergent 
series. 

For  the  second  case  Ave  have  the  sequence  of  coefficients 

(»^-l)(//^+2) 

(ni  -  1)  (m  -  3)  (m  +  2)  (m  +  4) 
a,  = 5i  a, 

(m  -  1)  (//^  -  3)  (m  -  5)  (>n  +  2)  (^^^  +  4)  (m  +  6)    _     ^^ 

7  -  -  I 

Let  us  take  a^ ,  which  is  arbitrary,  as  1.     Then  z  =  q,„(x)  where 

^       r-         (m-l)(m  +  2)    ^   ,   (m-1)  (n> -S)  (m +2)  (>n +i)    ^  n 

q,n(^)  =  |_a- 3! ^-   +  5l  •'•  J  (^> 

is  a  solution  of  Legendre's  Equation  if  </„  (x)  is  a  finite  sum  or  a  convergent 
series. 


Chap.  1.] 


LEGENDliE  S    EQUATION. 


21 


If  m  is  a  positive  even  whole  number, /»,„  (a')  will  terminate  with  the  term 
containing  ic'",  and  is  easily  seen  to  be  identical  with 


r  {1,1  + 1) 


[v.  Art.  9  (9)] 


For  all  other  values  of  m,  p^  (x)  is  a  series. 

The  ratio  of  the  (ji  +  l)st  term  of  ^j,„  (x)  to  the  nth,  when  vi  is  not  a  posi- 
tive even  integer,  is 


On  _  0  _ 


(271 


m)  (2?i  —  1  +  vi) 


(2?i  —  1)  (271) 

Its  limiting  value,  as  n  is   increased,  is  x^,  and  the   series   is  therefore  con- 
vergent if  —  1  <  a-  <  1.     It  is  divergent  for  all  other  values  of  x. 

If  m  is  a  positive  odd  whole  number  y,„  (.r)  will  terminate  with  the  term 
containing  x"\  and  is  easily  seen  to  be  identical  with 

+  1^ 


r  (m.  + 1 ) 

For  all  other  values  of  m,  q„^(x)  is  a  series,  and  can  be   shown   to   be   con- 
vergent if  —  1  <  .X'  <  1,  and  divergent  for  all  other  values  of  x. 


Ap,„(x)  +  Bq„^(.r) 


i.^) 


is  the  general  solution  of   Legendre's  Equation  if  —  1  <  x  <  1,  no  matter 
what  the  value  of  m.     From  Art.  13  (c)  it  follows  that 


V=  i'"p^  (cos  6) 
V=r"'q„^  (COS  6) 


-;^^   y,„(C0S^) 


(7) 


are  particular  solutions  of 


vD'^  (r  V)  -f  -^-p.  Be  (sin  6  I)g  V) 


0, 


no  matter  what  the  vakie  of  m,  provided  cos  0  is  neither  one  nor  minus  one. 

In  the  work  we  shall  have  to  do  with  Laplace's  and  Legendre's  Equations, 
it  is  generally  possible  to  restrict  ??i  to  being  a  positive  integer,  and  hereafter 
we  shall  usually  confine  our  attention  to  that  case. 


22  INTKODUCTION.  [Art.  16. 

With  this  understanding  let  us  return  to  (3),  which  may  be  rewritten 

(m  —  n)  (in  +  n  +  1) 

««  +  2  =  («  +  !)(« +  2)       ""-  ■ 

If  ««  +  2  —  0,  then  r?,^  _^  ^  =  0,    a^^^.  =  0  ,    &c. ; 

but  «^w  +  2  =^  ^    if    ''  =^  ^'^)    or    ?i  =^  —  m  —  1. 

If  in  (3)  we  begin  with  n  =  ni  —  2,  we  get  the  sequence  of  coefficients  already 
obtained  in  Art.  9,  and  we  have  z  =  P^(x),  where 

{2m  —  1)  {2m  —  3)  •  •  •  1  T  m{m.  —  1) 


1)- 


•]  .       (8) 


(2;.-l)(2».-3)  •••!  r      _   K 
^7»(,^;—  „^!  |_^         2(2 

??i(m  —  1)  (??i  —  2)  (7?i  —  3) 
^        2.4.  (2 /;i  —  l)(2»i  — 3)      ^™ 

■m(rti  —  1)  {m  —  2)  (7>i  —  3)  {m  —  4)  (??i  —  5)      _ 
2.4.6.(2  m  —  1)  (2  7m  —  3)  (2  m  —  5)         ^"^    ^"^ 

as  a  particular  solution  of  Legendre's  Equation. 
If,  however,  we  begin  with  w  =  —  m  —  3,  we  have 

{m  +  l){m  +  2) 
^-m-3—         2(2  VI  +  3)        *- '"  - 1 

{m  +  l)(m  +  2)(m+3){m  +  4:) 
^-m-6—  2A.(2m  +  3)  {2m  +  5)  ^-™-i 

(»^  +  1)  {m  +  2)  (^m  +  3)  (»^  +  4)  {m  +  5)  (».  +  6) 
a_^_7  —  2.4.6. (2 /M  +  3)  (2m  +  5)  (2»^  +  7)  «'-m-i »    «c. 

«_m_i  niay  be  taken  at  pleasure,  and  is  usually  taken  as  i   q  k  .  .  .  /9. ,  _l  i y 
and   z  =  Q,n{x)  where 

^    ,,_ !^i! r     1  {m  +  l){m  +  2)      1 

fj^m  C-^;  —  ^2m  +  1)  (2//^  —  1)  •  •  •  1  L  a-'"  +  ^  "^       2.  (2m  +  3)        a;'"  +  ^ 

•  (».  +  l)(m  +  2)(m+3)(m+4)       1  n 

"*"  2.4.(27?^  +  3)  (2m  +  5)  a;-  +  ^  ^  J  ^^'' 

is  a  second  particular  solution  of  Legendre's  Equation,  provided  the  series  is 
convergent.      Qm{x)  is  called  a  Surface  Zonal  Harmonic  of  the  second  kind. 


Chap.  I.]  ZONAL    HARMONICS.  23 

It  is   easily  seen  to  be   convergent  if  x-  <  —  1    or  a-  >  1,   and   divergent   if 
-1<  a-  <  1. 

Hence  if  m  is  a  positive  integer, 

z=AP,„{.v)  +  BQ,„{x)  (10) 

is  the  general  solution  of  Legendre's  Equation  if  x  <  —  1  or  x  >  1. 
We  have  seen  that  for  —  1  <  ,r  <  1 

-"  r  (m  +  1) 

2"'[r(,>i)] 

if  m  is  an  eveu  integer,  and 

?i^'  r  ill,  +  1) 

if  rn  is  an  odd  integer. 

If  now  we  define  (),„  (.r)  as  follows  when  —  1  <  ./•  <  1 

"±=1       r  ( w  +  1) 

Q^  (x)  =  (-  1)  -^  —--4- ^,  Pm  (•'•)  (13) 


if  ?«  IS  an  odd  integer,  and 


2'"[r(f +  1)]' 


-    .     r  (m  -f-  1) 


..-.[r(^)J 


if  m  is  an  even  integer,  then  (10)  will  be  the  general  solution  of  Legendre's 
Equation  if  m  is  a  positive  integer  when  —  1  <  a;  <  1,  as  well  as  when  x  <  —  1 
or  X  >  1. 

17.     Let  us  last  consider  the  equation 

(r-z       1  dz        /  m-\ 

l^^xdx+(}--¥)^  =  '  (1> 

which  is  known  as  Bessel's  Equation,  and  which  reduces  to  (8)  Art.  11, 
that  is,  to 

d^z        1    dz  _ 

dx~        X  dx       "^ 

■when  ?n  =  0;*  (1)  can  be  simplified  by  a  change  of  the  dependent  variable. 

*  This  equation  was  first  studied  by  Fourier  in  considering  tlie  cooling  of  a  cylinder.     \Ve 
shall  designate  it  as  "Fourier's  Equation." 


24  INTKODUCTION.  [Art.  17. 

Let  z  =^  cc"'  V  and  we  get 

cPv       2m +  1  dv 

s?  +  —r- &:  +  "  =  »  (2) 

to  determine  v. 

Assume  <;  =  2  a^x",  and  substitute  in  (2).     We  get 

2  [?i(2  m  -\-  n)a^x^^~'-^  +  «„a'"]  ^  0; 

whence  o„_2  =  —  n(2  m  +  n)a„. 

If  we  begin  with    n  =  0,    then    a„_2  =  0,    a„_^  =  0,    &c.,  and  we  have  the 
set  of  values 


^2  = 


2(2;// +2)-       2%m  +  l) 


'*  ~  2A(2m  +  2)(2m  +  4)  ~  2\2\{m  +  \){m  +  2) 


2.4.6(2?M  +  2)(2;m  +  4)(27>i  +  6)  2^3  !(m  +  l)(»i  +  2)(m  +  3) 


whence         z  =  a,x-^  [l  -  r^^;^!^^  +  2*.2!(m +%0"  +  2) 


+ 


2«3!(w  +1)(»/  +  2)(;m  +  3) 


]  (3) 


is  a  solution  of  Bessel's  Equation,  a^  is  usually  taken  as    -^^ — ^   if  m  is  a  pos- 
itive integer,  or  as    <^,„  -p  , — aTT^    i^  ^'^  is  unrestricted  in  value,  and  the  second 

member  of  (3)  is  represented  by  t/^(a;)  and  is  called  a  Bessel's  Function  of  the 
With  order,  or  a  Cylindrical  Harmo7iic  of  the  ?«th  order. 

If   m  =  0  ,    Jm{x)    becomes  Jo{x)  and  is  the  value  of  z  obtained  in  Art.  11 
as  the  solution  of  equation  (8)  of  that  article. 

If  in  equation  (1)  we  substitute  a--"'/?  in  place  of  x"\v  for  z,  we  get  in  place 
of  (2)  the  equation 

dh^        1  —2m  f/r 


dx'^  x        dx' 


and  in  place  of  (3) 


r  x 


V    2^2!(1—  m)(2  —  m) 
2«.3!(1  —  m)C2  —  m){3-  n>)  +  '  '  '  J  (^^ 


Chap.  T.]  BESSEL's    EQUATION.  25 

If  Oq  is  taken  equal  to  the  second  member  of  (4)  is  the  same 

U  " '"  r  (1  —  m)  ^ 

function    of   —  in    and  x  that  J,,,  (.r)  is  of   +  m  and  x  and  may  be  written 
Therefore  z  =  AJ,„  (,r)  +  BJ^,^  (.r)  (5) 

is  the  general  solution  of  (1)  unless  '/,„(»')  and  J_,,^(^x)  should  prove  not  to  be 
independent. 

It  is  easily  seen  that  Avhen  m  =  0 ,  J_  „,  (x)  and  J,„  (x)  become  identical 
and  (5)  reduces  to 

r:  =  (A+I-r)J,(x) 

and  contains  but  a  single  arbitrary  constant  and  is  not  the  general  solution  of 
rourier's  Equation  (8)  Art.  (11). 

It  can  be  shown  that  (/_„j(.r)  =  (— l)'"cr,„(.T)  whenever  m  is  an  integer, 
and  consequently  that  the  solution  (5)  is  general  only  when  m  if  real  is  frac- 
tional or  incommensurable. 

The  general  solution  for  the  important  case  where  m  ==  0  is,  however,  easily 
obtained.  Let  F(m,x)  be  the  value  which  the  second  member  of  (3)  assumes 
when  rto  =  1 ;  then  the  value  which  the  second  member  of  (4)  assumes 
when  ((q  =  1  will  be  F( —  w,a-),  and  it  has  been  shown  that  z  =  F(7n,x)  and 
z  =  F( —  vi,x)  are  solutions  of  Bessel's  Equation;  z  =  F(in,x)  —  F(^ —  m,x) 
is,  then,  a  solution,  as  is  also 

F{m.,x)  —  F{—  ni,x) 

z  = -^ ■ >  (O; 

1)11  ^ 

but  the  limiting  value  which  — ^   ■'        ^ — ^^ — — -—  approaches  as  m  approaches 

zero  is  [A/i^("^v'')]/»=ii  and  consequently 

z  =  lD„,F{in,x)l,^,  (7) 

is  a  solution  of  the  equation 

d:'z       1  dz 

d^'r-j^  +  ^^O,  (8) 

and  the  general  solution  of  (8)  is 

z  =  A  J,(x)  +  B  [A„  F  (m,  .t)],,^„  . 


i^0.,..)  =  .-^-"{i-2^T)  + 


2\m  +  1)    '   2\2l(m  +  l)(w  +  2) 

^ 

2V3l(m.+l)(Ni-{-2)(m  +3)  +  '  ' 


26  INTRODUCTION.  [Art.  17. 

D,„F{m.,  ,x)  =  ,r"'  log  ..  [l  -  ..^  J+ ^^  +  2^2 !(»..  +  !)(..  + 2) ] 

+  ■'^"'  A„,  [_1  -  2%i^-+T)  +  2''.2!(;m  +  1)(w  +2)  ^  J  • 

The  general  term  of  tlie  last  parenthesis  can  be  written 

(-  1)^- 


'2-''.kl{m  +  l){m.  +  2)  ■  ■  •  (m  +  k) 
and  its  partial  derivative  with  respect  to  m  is 

1 


(-   1)'  o:«-7-.  B 


2*.  /.•!  ■^">  (m  +  1)(^»^  +  2)  •  •  •  (;m  -+-  A-) 

1 

log ^j— ^— j-^  =  —  flog  (la  +  1)  +  log  (III  +2)  +  •  •  • 

(ill  -f-  l)(i7i  +  2)  .  .  .  (?»-  -\-  k)  L     &  V       I      /    I       »  V       I     /    I 

+  log  (m  +  ;.■) J  . 

Take  the  D,„  of  both  members  and  we  have 

1 

^"^  (w.  +  1)  {iii  +  2)  •  •  •  (ill  +  A") 

~       (III  +  l)(«i  +2)  •  •  •  (III  +  k)   \_iii  +  1  ^  »^  +  2  ^  //;  +  /.■  J 

^'"  |_-^  ~  2-^(»^  +  1)  +  2*.2!(w  +  l)(w  +  2}  ~  2-^.3 !(;;/  +l)(w+  2)(»?  +3) 

-1  _  r^      1         j^ 1  r    1  in 

+  ■  ■  ■  J  ~  2^  (»^  +  1/  " iK2!  (/M  +  1)(///  +2)  L^^^Tl  ^  ^^H^' J 

.T«  1  fill  1        ~| 

and  Ave  have 

x^      1  ,t4      /I      1\  .r^      /I       1      1\ 

[X>^i^(m, a;)] „,^ 0  =  ^o(-^')  log  .r  +  2^!)2 1  ~ 2V2  ij^  Vl  ^  2/  +  2W!p  U  +  2  +  3/ 

x'    n     1     1     i\ 

~  2«(4  !7  U  +  2  +  3  +  4/  "*  ' 

and  z=A Jo (x)  +  B A'o (a-)  ,  (9) 

X,  5C*       /I        1\  x^       /I       1        1\ 

where     Ko  (x)  =  J^  (x)  log  x  +  02  "  2V2 !?  U  "^  2/  +  28(3 !)-  U  "^  2  +  3/ 

.T^        /I  1  1  1\ 

-2^J404l+2  +  3+4)  +  ---  (^^^ 

is  the  general  solution  of  Fourier's  Equation  (8). 

Kq(x)  is  known  as  a  BesseVs  Function  of  the  Second  Kind. 


Chap.  I.]  GENERAL    SOLUTION.  27 

18.  It  is  worth  while  to  conlirni  the  results  of  the  last  few  articles  by- 
getting  the  general  solutions  of  the  equations  in  question  by  a  different  and 
familiar  method. 

The  general  solution  of  any  ordinary  linear  differential  equation  of  the 
second  order  can  be  obtained  when  a  particular  solution  of  the  equation  has 
been  found  [v.  Int.  Cal.  p.  321,  §  24  (r?,)]. 

The  most  general  form  of  a  homogeneous  (uxlinary  linear  differential  equa- 
tion of  the  second  order  is 

where  P  and  Q  are  functions  of  x.     Suppose  that 

//  =  "  (2) 

is  a  particular  solution  of  (1).     Substitute  y  :=  nz  in  (1)  and  we  get 
(]:'z       /    dv  \  dz 

dz 
Call      -j~  =  z'.     Then  (3)  becomes 

a  differential  equation  of  the  first  order  in  which  the  variables  can  be  sepa- 
rated.  Multiply  by  dx  and  divide  by  vz'  and  (4)  reduces  to 

dz'  dr 

-^  +  2  ---  +  Pdx  =  0  . 

Integrate  and  we  have 

log  z'  +  log  v'  +  fPdx  =  C 

or  z'r'  =  e^'-  f^''^  =  Be-f^'^^  , 


Iz         ^   e-J''"- 

fPdx 


dx 


and 


^^A+BJ--^-         dx  ; 

,j  =  >,  (a  +  Bf'^\lx^  (5) 

is  the  general  solution  of  (1),  the  only  arbitrary  constants  in  the  second  mem- 
ber of  (5)  being  those  explicitly  written,  namely,  A  and  B. 
(a)     Apply  this  formula  to  (1)  Art.  14, 

£  +  «'^  =  0;  (1) 


28  INTEODUCTIOX.  [Akt.  18. 

given :    x  =  cos  ax,    as  a  particular  solution.     Substituting  in   (5)   we  have 
since  F  =  0 


z  =^  cos  ax 


\  J    cos^  a.r/ 

/  B  \ 

I  -4  H tan  ax  I 


=  A  cos  ax  +  7>i  sin  ax  ,  (2) 

as  the  general  solution  of  (1),  and  this  agrees  perfectly  with  (5)  Art.  14. 

(b)  Take  equation  (1)  Art.  15. 

d-'z  dz 

^r7^  +  2^^-"K'-  +  l>  =  0;  '(1) 

given:  z  =  .r™,    as  a  particular  solution. 

2       r  -rpdx      1 

Here   P  =  -  ,    j    Pdx  =  2  log  x  =  log  x'- ,    and    e  '        =-.,.     Hence  by  (5) 

that  is  z  —  Ax"^  +  rr^^i  (2) 

is  the  general  solution  of  (1),  and  agrees  with  (2)  Art.  15. 

(c)  Take  Legendre's  Equation,  (2)  Art.  16. 

<^^  ~  ^'^  rfe^2  -  2a;  ^  +  7/^(m.  +  1)".  =  0  ;  (1) 

given :    z  =  P,,^  (.r) ,    as  a  particidar  solution. 

Here     P  =  j^z:'"^  ,     J  Pdx  =  log  (1  -  x^)  ,     and     e-/^^'-  =  j^— a* 

Hence  by  (5)  .  =  P,„(.)  (.4  +  p J^-^-_|^-^^.^)  (2) 

is  the  general  solution  of  (1)  and  must  agree  Avith  (11)  Art.  16,  if  m  is  an 
integer,  and  therefore 

where  C  is  as  yet  undetermined,  and  no  constant  term  is  to  be  understood  with 
the  integral  in  the  second  member. 

(d)  Take  Bessel's  Equation,  (1)  Art.  17. 

d'^z       Idz        /         m\ 

^^  +  ^^^  +  (l-^>^  =  ^'  (1) 

given :    z  ^  J^(x)  ,    as  a  particular  solution. 


Chap.   I.]  GENERAL    SOLUTION.  29 

1  />  />  1 

Here        ^  =  ^  >      I    ^^^^  =  log  *'  »     ^^^^     e"^  ^''^  =  -  .     Hence  by  (5) 

.,=^„.(.)(..i+/./-^,)  (2) 

is  the  general  solution  of  Bessel's  Equation. 
If  m  =  0  (2)  becomes 

^  =  M-)(-i+^f-^;^:)  (3) 

and  must  agree  with  (9)  Art.  17.     Therefore 

if.(x)  =  C^.(.)Jjp^^,  (4) 

where   C  is   at  present  undetermined,  and  no   constant  term  is  to  be  taken 
with  the  integral. 

The  first  considerable  subject  suggested  by  the  problems  which  we  have 
taken  up  in  this  introductory  chapter  is  that  of  development  in  Trigonometric 
Series  (v.  Arts.  7  and  8). 


CHAPTER   II. 

DEVELOPMENT    IN    TRIGONOMETRIC    SERIES. 

19.  We  have  seen  in  Chapter  I.  that  it  is  sometimes  important  to  be  able 
to  express  a  given  function  of  a  variable  cc,  in  terms  of  the  sines  or  of  the 
cosines  of  multiples  of  x.  The  problem  in  its  general  form  was  first  solved 
by  Fourier  in  his  "Analytic  Theory  of  Heat"  (1822),  and  its  solution  plays  a 
very  important  part  in  most  branches  of  modern  Physics.  Series  involving 
only  sines  and   cosines  of  whole  multiples  of  x,  that  is  series   of  the  form 

/>o  +  hi  cos  X  -\-  1)2  cos  2x  -\-  ■  •  •  -\-  (ii  sin  x  +  «2  sin  2x-\-  •  •  • 
are  generally  known  as  Fourier's  series. 

Let  us  endeavor  to  develop  a  given  function  of  x  in  terms  of  sin  x,  sin  2x, 
sin  'dx,  &c.,  in  such  a  way  that  the  function  and  the  series  shall  be  equal  for 
all  values  of  x  between  cc  =  0  and  ic  =  tt. 

To  fix  our  ideas  let  us  suppose  that  Ave  have  a  curve, 

given,  and  that  we  wish  to  form  the  equation, 

ij  =  (ii  sin  X  -\-  a^  sin  2x  +  a^  sin  3x  -\-  •  ■  • , 
of  a  curve  which  shall  coincide  Avith  so  much  of  the  given  curve  as  lies  between 
the  points  corresponding  to  a-  =  0  and  a-  =  tt. 
It  is  clear  that  in  the  equation 

y  =  r^i  sin  X  (1) 

a  I  may  be  determined  so  that  the  curve  represented  shall  pass  through  any 
given  point.  For  if  we  substitute  in  (1)  the  coordinates  of  the  point  in  ques- 
tion we  shall  liave  an  equation  of  the  first  degree  in  which  a^  is  the  only 
unknown  quantity  and  which  will  therefore  give  us  one  and  only  one  value 
for  a-i . 

In  like  manner    the  curve 

y  =.  a  I  sin  X  +  (to,  sin  2a! 
may  be  made  to  pass  through  any  two  arbitrarily  chosen  points  whose  abscissas 
lie  between  0  and  tt  provided  that  the  abscissas  are  not  equal;  and 

ij  =.  Ki  sin  X  +  Go  sin  2x  +  (h  sin  3x  -\-  •  •  ■  -\-  a„  sin  nx 
may  be  made  to  pass  through  any  w  arbitrarily  chosen  points  whose  abscissas 
lie  between  0  and  ir  provided  as  before  that  their  abscissas  are  all  different. 

If,  then,  the  given  function  f(x)  is  of  such  a  character  that  for  each  value  of  x 
between  x  =  0  and  cc  =  tt  it  has  one  and  only  one  value,  and  if  between 
a;  =:  0  and  x  =  ir  it  is  finite  and  continuous,  or  if  discontinuous  has  only 
finite  discontinuities  (v.  Int.  Cal.  Art.  83,  p.  78),  the  coefficients  in 

y  =  a  I  sin  X  +  a^  sin  2x  -\-  a^  sin  3a;  ■{-•■•  +  ''«  sin  nx  (2) 


PRELIMINARY    STUDY    OF    A    FINITE    SUM.  81 

can  be  determined  so  tliat  the  curve  represented  by  (2)  will  pass  tlirougli  an}- 
II  arbitrarily  chosen  points  of  the  curve 

'I/  =  f(^)  (3) 

whose  abscissas  lie  between  0  and  tt  and  are  all  different,  and  these  coefficients 
will  have  but  one  set  of  values. 

For  the  sake  of  simplicity  suppose  that  the  n  points  are  so  chosen  that  their 
projections  on  the  axis  of  A' are  equidistant. 

TT 

Call  ,— iT'T  =  '^•^"  ;  then  the  coordinates  of  the  n  points  will  be  [A,x,/(zia;)], 
[2A.x',/(2Aa")],  [3Ax,/(3Aa-)],  •  •  •  [MAcc,/(?^Aa:)].  Substitute  them  in  (2)  and 
we  have 

/(Aa;)  =    (ii  sin  Ax  -\-    a^,  sin  2A.r  -\-    a.,  sin  3A;r  +  '  '  '  +    "«  sin  7iAx'^ 

/(2Aa-)  =  «i  sin  2Aa;  +    (1.2  sin  4A.i'  +    rtg  sin  6Ax  + h  "„  sin  2uXr  \ 

/(SAa-)  =  ('i  sin  3\x  +    >i.,  sin  6A.r  +    x.^  sin  9A.r  -\ \-  >i,,  sin  onAx  [  '' 

/(«  A.x)  =  a-^  sin  iiAx  -\-  a^  sin  2n\x  +  a^  sin  3»A./;  -[-•••+  "«  sin  ii^Ax,  j 

»j  equations  of  the  first  degree  to  determine  the  n  coefficients  a-^ ,  "^o,,  (tz,  •  •  -  a,, . 

Xot  only  can  equations  (4)  be  solved  in  theory,  but  they  can  be  actually 
solved  in  any  given  case  by  a  very  simple  and  ingenious  method  due  to 
Lagrange. 

Let  us  take  as  an  example  the  simple  problem  to  determine  the  coefficients 
cti,  (u,  «3,  cti,  and  rtg,  so  that 

//  =  r/i  sin  X  -\-  a.2  sin  2x  +  «3  sin  3x  -\-  u^  sin  4,<f  +  ((r,  sin  ox  (o) 

shall  pass  through  the  five  points  of  the  line 


77    27r  Stt    Att         -.   Stt      tt 

which  have  the  abscissas     77  >    ,—  ?  -,—  ?    .1- '  '^^it*-  -?r  >    'f  here  being  A.-r . 

b      ()  b       b  6        b  ® 

We  must  now  solve  the  equations 

TT  .        TT  .        27r  .        StT  .        47r  OTT 

—  :=  c/i  sin  TT   +  (lo  sin  -rr-  +  «3  sin  -jr  +  ''/4  sin  — ;-  -j-  «5  sin  -rr- 

27r  27r  47r  Gtt  Stt  10,t 

— -  =:  <^i  sin  -7,-   +  r?3  sin  -7 1-  ^^3  sin  -^   -\-  a^  sin  --  +  a^  sin  —.- 

37r  Stt  Qtt  .      Ott  127r  Iott  >  (61 

--  =  (^j  sin  -TT   +  r^.o  sin  -77-   +  ^3  sm  -r,-  +  11^  sm  — , [-  ^5  sm  -— .- 

47r  .     47r  .      Stt  .     127r  .     IOtt  .     207r 

-~  =  a^  sin  —p~  +  </2  sin  -7,-   +  "3  sin  —7. 1-  a^  sin  —7^ \-  a^  sin  — ,.— 

57r  Stt  .     IOtt   ,  .     Iott   ,  .20-77,  .     257r 

((i  sin  -^   +  (1,2  sm  — ,, 1-  (Iq  sm  -^ 1-  (1.^  sm  -7^ \-  a.,  sm  -.7 


32 


DEVELOPJNIENT    IN    TRIGONOMETRIC    SERIES. 


tArt.  20. 


Multiply  the  first  equation  by  2  sin  g  ,  the  second  by  2  sin  -^   ,   the  third 

by   2  sin  -^  ,   the  fourth  by   2  sin  -g-,    the   fifth  by   2  sin  -g-   and   add    the 

equations. 

The  coefficient  of  a.  is 


TT    .     27r    ,     ^     .      27r     .      47r    ,    ^     .     Sir 
2  sin  g  sm  -g-  +   2  sm  -jt  sm  "rr  +  2  sm  —  sm 


6 


6 


but 


Stt         IOtt 
+  2  sin  -7.--sin  -^; 


2  sin  TT  sin 


Stt 


67r    I    o     •     47r     .     Stt 

T  +  ^  >*■"  -g-  «■"  T 


&C. 


Hence  the  coefficient  of  (u  becomes 


TT  27r  Stt  47r  Stt 

cos  TT  +  cos  -77-  +  cos  -g-  +  cos  -g — h  cos 


Stt 


Gtt 


dir 


VItt 


6 

157r 
"6" 


(7) 


and  this  may  be  reduced  by  the  aid  of  an  important  Trigonometric  formula 
which  we  proceed  to  establish. 


20.     Lemma. 

cos  ^  +  cos  2 ^  +  cos  3 ^  +  •  •  •  +  cos  n6 


^       ^  sin(2«  +  1)  2 
919  \     ^ 


(1) 


For  let    *S'  =  cos  ^  +  cos  IB  +  cos  3^  +  •  •  °  +  cos  nQ    and  multiply  by  2  cos  ^ . 

2»S  cos  ^  =  2  cos^  ^  +  2  cos  ^  cos  2^  +  2  cos  ^  cos  3^  H H  2  cos  0  cos  nO 

=  1  +  cos  ^  +  cos  2^  H h  cos  (m  —  1)  ^ 

+  cos  20  +  cos  3^  +  cos  4^  H \-  cos  (ii  +  1)0 


=  2>S'  +  1  +  cos  {11  +  1)  ^  —  cos 


Hence 


S  = 


+ 


cos  nO  —  cos  {n  +  1) ^ 
2(1— cos  ^) 


^       ^  sin  (2«  +  1)2 

'^  "^  ~  2  "^  2 ~e 

sin  1^ 


Chap.  II.]  NUMERICAL    EXAMPLE.  33 

21.     Applying  (1)  Art.  20  to  (7)  Art.  19  the  coefficient  of  a^  reduces  to 

^37r 
l2 


IItt         .     337r 
—^-7v        sin- 


2 

sm  j2       2  sm  j-^- 

IItt 

12    -'^- 

TT       .        337r                  37r 
■  j2  '   ^^^cl   ^1^  =  37r  —  ^2 

refore 

sin(7r-f,) 

™(s^-f|)      J      J 

and  a.2  vanishes. 

In  like   manner  it   may  be  shown  that  the   coefficients  of  a^,  a^,  and  a^ 
vanish. 

The  coefficient  of  ^i  is 

2  sin^  5+2  sin-^  -J  +  2  sin'^  ^r  +  2  sin^  ^ 
o  b  (>  () 


=1+1+1+1 

+         1 

27r               47r               Gtt                Stt 

—  cos cos  — cos cos    -~ 

6                6                6                  6 

IOtt 
cos     g 

•      IIt^                         •     /..            TTX 
sm  ---                    sm  (  27r  —  -  J 

2  Sin  -                        2  sm  - 
6                                6 

=  6. 

The  first  member  of  the  final  equation  is 

2tt    .     TT  27r    .     2'ir  OTT    .     37r  47r    .     47r   ,    ^  Stt    .     57r       ^ 

-sm-+2--sm--  +  2--sm--  +  2-sm-,  +  2--sm--.    Hence 

k=o 

2  -KT^  kir    .     k.TT       TT 


-  2^  —  sm  —  =  -  (2  +  ^3)  =  2      approximately. 


If  we  multiply  the  first  equation  of  (6)  Art.  19  by  2  sin  -^  ,  the  second  by 

2  sin  ^  ,    the  third   by     2  sin  -^ ,    the  fourth  by    2  sin  ^ ,      the     fifth 

IOtt 
by    2  sin  -—  ,  add  and  reduce  as  before  we  shall  find 

t  =  5 
2    ^    /.-TT      .       2A-7r  TT      ,.,  ,    ^ 

i=i 


84  DEVELOPMENT    IN    TKIGONOMETllIC    SEKIES.  [Akt.  22. 

and  in  like  ananner  we  get 

/I- =5 
2   .r-V   A'TT      .       3A'7r  IT  ^  _ 


«.  =  iXT-T  =  f(2-^«)  =  «-l 


Therefore 


^  =  2  sin  X  —  0.9  sin  2.r  +  0.5  sin  3.r  —  0.3  sin  4a?  +  0.1  sin  5x         (1) 

TT     27r     37r 
cuts  the   curve    >/  =  oc.     at  the   live   points  whose  abscissas  are    tt  ,  -^ ,  -tt  > 

47r  ,  Stt 

22.  The  equations  (4)  Art.  19  can  be  solved  by  exactly  the  same  device. 
To  find  any  coefiicient  «^  multiply  the  first  equation  by  2  sin  m^x .  the 
second  by  2  sin  27nAx,  the  third  by  2  sin  3m^x,  &c.  and  add. 

The  coefiicient  of  any  other  a  as  o^.  in  the  resulting  equation  will  be 

2  sin  kAx  sin  m\x  +  2  sin  2kL^x  sin  2/HA.r  +  2  sin  3/l'Acc  sin  3mAx  -{-■•■ 
-\-  2  sin  nk\x  sin  nm^x 

=  cos(/ft  —  A:)  Ax + cos  2  (//i  —  A-)A.T  +  cos3(;yi  — A')A.ifH \-cosn(m  —  k)Ax 

—  cos(v»  +  7.")  Ax— cos  2(m  +  k)\x  —  cos  3(?m  +  A-)  Ax cos  /i(Hi  +  '''■)  Ax 


by  (1)  Art.  20. 


n  -{-  1  —  -     and  (/i  +  l)Ax  =  tt  . 
^Hence  the  coefficient  of  a^.  may  be  written 

sin     (»i  -  k)7r  - -^-^           sm  (m  +  /(>  -  ^ — -^ — 

2  sm  ^^ o''  ^  sni  ^ ^ — 


2)1  4-1 
sin  ^y —  (/ft  —  A')  Ax 

2n  4-  1 
sin —  (m  +  /.■)Ax 

2  sm  ^        ^ 

2«  +  1  _ 

^    .     (>ft  +  A-)Ax 
2  sm  ^        2 

,    _L    1                            o^irl           /»    _1_    1\ 

but  this  is  equal  to   ^  —  ;^  or  —  -  +  -   according  as   m  —  k  is   odd 
and  so  is  zero  in  either  case. 


or  even 


Chap.  II. ]  DETERMINATION   OF    COEFFICIENTS.  35 

The  coefficient  of  a„,  will  be 

2  sin"^  )n.\x  +  2  sin^  2?/iAx  +  2  sin^  3m^x  +  •  •  •  +  2  sin^  nm^x 

1         +         1         +         1         +•••+         1 
—  cos  2wAu?  —  cos  4mA.x  —  cos  6m/\x  —  ■  •  •  —  cos  2)im/lx 

=  ■„  +  I  -  ^'"f'!  +  ^>'"^" ,  by  (1)  Art.  20. 

'    2  2  sill  mi\x  -^  ^  ^ 

But  (2n  +  l)mA«  =  2m(??,  +  l)Aa;  —  mA.'K  =  2m7r  — mAx, 

^        p    ^  sin  (2n-\-  l)))iAx sin  (2m7r  —  7ftAa;)  1 

2  sin  /mA.t  2  sin  7nAx  2  ' 

and  the  coefficient  of  a^  is     n  -\-  1 . 

The  first  member  of  our  final  equation  will  be 

2  "^/(kiXx)  sin  km/^x  . 

2      '^" 
*™  ^  ^7+1  2y  /('^^^)  Sill  '^'"^'^^  ;  (1) 

=  «i  sin  X  +  f/2  sin  2x  -\-  •  •  •  -^  a„  sin  ?ia; ,  (2) 


Hence 


and  the  curve 


where  the  coefficients  are  given  by  (1)  will  pass  through  the  n  points  of  the 

TT 

curve    ?/  =f(x)    whose  abscissas  are  Ace,  2Aaj,  SAx,  ■  •  •  nAx.  Ace  being —p- . 

It  should  be  noted  that  since  the  n  equations  (4)  Art.  19  are  all  of  the  first 
degree  there  will  exist  only  one  set  of  values  for  the  n  quantities  «i,  a,,  03, 
•  •  •  a„  that  can  satisfy  these  equations.  Consequently  the  solution  which  we 
have  obtained  is  the  only  solution  possible. 

23.  The  result  just  obtained  obviously  holds  good  no  matter  how  great  a 
value  of  n  may  be  taken. 

If  now  we  suppose  n  indefinitely  increased  the  two  curves  (2)  Art.  22  and 
y  =  f(x)  will  come  nearer  and  nearer  to  coinciding  throughout  the  whole  of 
their  portions  between  .r  =  0  and  cc  =  tt  ,  and  consequently  the  limiting 
form  that  equation  (2)  Art.  22  approaches  as  n  is  indefinitely  increased  will 
represent  a  curve  absolutely  coinciding  between  the  values  of  as  in  question 
with    y=f(x). 


36  DEVELOPMENT    IN    TRIGONOMETltlC    SERIES.  [Aht.  •2i. 

Let  us  see  what  limiting  value  (t,,^  approaches  as  n  is  indefinitely  increased. 

a^.^^  =  — ^  y  /(A-Aic)  sin  kmAx  (1)  Art.  22. 

2A.T    -r-^ 

^ '-  2_,  /(A'-^-^O  sin  Jcm/lx 

1  =  1 

=  -  [/(Aa-)  sin  7rtA.-r.A.r  +  /(2Aa:;)  sin  2»^Aa:'.A./'H {-/(iiAa')  sin  7im\x.Xr 

_  2  r  f ( A.r)  sin  mXr.  A.r  +  /'(2 A.r)  sin  2 /». A,i'.  A.r  H -j 

^|_  +/('"'  —  A.r)sin»;(7r  — A.'').A./'J 

since  A.r  =  — ; — -  • 
n  +  1 

As  n  is  increased  indefinitely  Aa.  approaches  zero  as  a  limit.     Hence  the 
limiting  value  of  «,„  as  n  increases  indefinitely  is 

2    limit    r/(Ax)  sin  mAx.Ax  +/(2A./-)  sin 2«iA.r. A.r  -\ ~|  ^ 

TT  Aa'  =  oL  +fi'^  —  -^•'•)  sin  ini-ir  —  A./-).A,i  J 

^a-)  sin  mx.dx .         [v.  Int.  Cal.  Arts.  SO,  81.] 


^/'^^ 


Hence  /(cc)  =  «i  sin  .r  +  a^  sin  2x  +  ^^3  sin  3.r  +  •  •  •,  (2) 

Avhere  any  coefficient  a,„  is  given  by  the  formula 

a„^  =  -    j  /(^O  sin  mx.dx  ,  (3) 

is  a  true  development  of  f(x)  for  all  values  of  x  between  x  =  0  and  .*•  =  tt 
provided  that  the  series  (2)  is  converr/ent,  for  it  is  in  that  case  only  that  we  can 
assume  that  the  limiting  value  of  the  second  member  of  (2)  Art.  22  can  be  ob- 
tained by  adding  the  limiting  values  of  the  several  terms. 

When  X  =  0  and  when  x  =  it  every  term  in  the  second  member  of  (2) 
is  zero,  and  the  second  member  is  zero  and  will  not  be  equal  to  f(x)  unless  f(x) 
is  itself  zero  when  x  =  0  and  x  =  tt  ;  but  even  when  f(x)  is  not  zero  for 
X  ^  0  and  a;  =  tt  the  development  given  above  holds  good  for  any  value 
of  x  between  zero  and  tt  no  matter  how  near  it  may  be  taken  to  either  of  these 
values. 

24.  Instead  of  actually  performing  the  elimination  in  e.juations  (4)  Art. 
19  and  getting  a  formula  for  «,„  in  terms  of  n,  and  then  letting  n  increase 
indefinitely,  we  might  have  saved  labor  by  the  following  method. 

*  We  shall  vise  the  sign  ==  for  approaches.     Ax  =  0    is  read  Ax  approaches  zero. 


Chap.  II.J  ABRIDGED  METHOD.  37 

Return  to  equations  (4)  Art.  19  and  multiply  the  first  by  Aic  sin  inb>.Xy 
the  second  by  Aa;  sin  2mA.r,  and  so  on,  that  is  multiply  each  equation  by  ^x 
times  the  coefficient  of  a^  in  that  equation,  and  then  add  the  equations. 

We  get  as  the  coefficient  of  % 

sin  k\x  sin  m^x.  Aj:-  +  sin  2^-Ax-  sin  'ImH^^x.  A.r  -j -|-  sin  nk^x  sin  nniL^x.  Ax . 

Let  us  find  its  limiting  value  as  n  is  indefinitely  increased.  It  may  be 
written,  since  (/i  +  1)  A»'  =  tt  , 


limit    rsinA;Aa:;sin7/iAa:'.  A;j"  +  sin2A^A,Tsin2w.Aa;.  Aa--] ~1 

L.X'  =  0  L  +  sin  ^  (tt  —  Aa;)  sin  m  (.tt  —  \x) .  Ax  J 

=    I  sin  kx  sin  vix.  dx  ; 


but  I  sin  kx  sin  mx.dx  =  i  |   [cos  (m  —  k^x  —  cos  {^n  +  k>)x\dx 

(I  0 

=  0  if  w,  and  k  are  not  equal. 
The  coefficient  of  a,„  is 

Aa;(sin-  m!^x  +  sin"^  2?MAa-  +  sin'-^  ?>mAx  +  •  •  •  +  sin'^  v^mA,-;c) 

Its  limiting  value 


limit 

Ax 


_  ,.      sin'^  y«,Aa:;.A;r  +  sin- 2mAa:;.A.r  +  ■  •  •  +  sin2?«,(7r  —  A,r)A.:f 


sin'^  mx.dx  =  —  • 

The  first  member  is 
/"(Aa')sin  7/?A./'.A.r  H-/(2Aa-)  sin  2mAx.Ax  +  ■  ■  ■  +/(wAa;)  sin  mnAx.Ax 
and  its  limiting  value  is 

I  f(oii)  sin  mx.dx  . 

0 

Hence  the  limiting  form  approached  by  the  final  equation  as  n  is  increased  is 
^  f{x)  sii 


sin  mx.dx  =  -  a. 


//(.X-) 


Whence  a,,,  ^  -    |    f(x)  sin  mx.dx  as  before. 


This  method  is  practically  the  same  as  multiplying  the  equation 

f(x)  =  cii  sin  X  -\-  ^2  sin  2x  +  a^  sin  3x  +  '  '  '  (1) 

by  sin  mx.  dx  and  integrating  both  members  from  zero  to  tt  . 


38  DEVELOPMENT    IN    TRIGONOMETRIC    SERIES.  [Art.  -2.5. 

It  is  exceedingly  important  to  realize  that  the  short  method  of  determining 
any  coefficient  cf„^  of  the  series  (1)  which  has  just  been  described  in  the  itali- 
cized paragraph,  is  essentially  the  same  as  that  of  obtaining  «,„  by  actual 
elimination  from  the  equations  (4)  Art.  19,  and  then  supposing  7i  to  increase 
indefinitely,  thus  making  the  curves  (3)  Art.  19  and  (2)  Art.  19  absolutely 
coincide  between  the  values  of  x  which  are  taken  as  the  limits  of  the 
definite  integration. 

25.  We  see,  then,  that  any  function  of  x  Avhich  is  single-valued,  finite,  and 
continuous  between  x  =  0  and  x  =  tt,  or  if  discontinuous  has  only  finite 
discontinuities  each  of  which  is  preceded  and  succeeded  by  continuous  por- 
tions, can  probably  be  developed  into  a  series  of  the  form 

f(x)  =  a^  sin  x  -\-  0.2  sin  2x  +  r/.,  sin  3.t  +  •  •  •  (1) 

where  «„,  =  -   |  f(x)  sin  mx.dx  —  -    i  f(a)  sin  tna.da  ;  (2) 

and  the  series  and  the  function  will  be  identical  for  all  values  of  x  between 
X  =  0  and  cr  =  tt,  not  including  the  values  .t  =  0  and  x  =  tt  unless 
the  given  function  is  equal  to  zero  for  those  values. 

An  elaborate  investigation  of  the  question  of  the  convergence  of  the  series 
(1),  for  which  we  have  not  space,  entirely  confirms  the  result  formulated 
above  *  and  shows  in  addition  that  at  a  point  of  finite  discontinuity  the  series 
has  a  value  equal  to  half  the  sum  of  the  two  values  which  the  function 
approaches  as  we  approach  the  point  in  question  from  opposite  sides. 

The  investigation  which  we  have  made  in  the  preceding  sections  establishes 
the  fact  that  the  curve  represented  by  ?/  —  f(x)  need  not  follow  the  same 
mathematical  law  throughout  its  length,  but  may  be  made  up  of  portions  of 
entirely  different  curves.  For  example,  a  broken  line  or  a  locus  consisting  of 
finite  parts  of  several  different  and  disconnected  straight  lines  can  be 
represented  perfectly  well  by  //  =  a  sine  series. 

26.  Let  us  obtain  a  few  sine  developments. 

(a)     Let  fi^')^'^.  (1) 

We  have  x  =  a^  sin  x  +  f'2  sin  2.r  +  '''3  sin  3a:;  +  •  •  •  (2) 


'here 


TT 


j  X  sin  mx.dx  (3) 


*  Provided  the  function  has  not  an  infinite  number  of  maxima  and  minima  in  tlie  neioh- 
borhood  of  a  point,     v.  Arts.  37-38. 


Chap.  IL] 


EXAMPLES    OF    SINE    SERIES. 
I  X  sin  m.x.dx  =  --  (sin  mx  —  mx  cos  ?»,r), 

(-IV'TT 


and 

{h)     Let 


'<-. 


/x  sin  mx.  dx 
III 

sin  X       sin  2x    ,    sin  3.r       sin  4,7- 


+ 


m 

9 


a^  ^  —    I  sin  mx.dx  ; 


(4) 
(1) 

(2) 


/ 


/^^ 


sin  mx.dx  =  — 

1 


cos  VIX 
III 


sin  ma;.  (7a-  =  -  (l  —  eos  iinr) 


[l-(-l)-] 


0  if  m  is  even 


=  —  if  VI  is  odd. 


Hence 


4  /sin  X    ,    sin  3.r    ,    sin  5a'    ,    sin  7x 


(3) 


It  is  to  be  noticed  that  (3)  gives  at  once  a  sine  development  for  any  constant 
c.     It  is, 


4c  /sin  X       sin  3a'       sin  5x  \ 


(4) 


If  we  substitute  a-  =  —  in  (4)  (rr)  or  (3)  (b)  we  get  a  familiar  result,  namely 

(5) 


4        13       5       7, 


a  formula  usually  derived  by  substituting     x  =  1     in  the  power  series  for 
tan- la',     (v.  Dif.  Cal.  Art.  135.) 

(4)  («)  does  not  hold  good  when   x  =  tt,   and  (3)  (fj)  fails  when   x  =  0   and 
when   X  =  TT,    for  in  all  these  cases  the  series  reduces  to  zero. 

(c)    Let  f(x)  =  X  from  x  =  0  to  .'■  ■ 


and    f(x) 


from  a- 


to  .r  =  TT  • 


That  is,  let  ^  =  f(x)  represent  the  broken 
line  in  the  figure. 

As    the     mathematical    expression   for 
f(x)  is  different  in  the  two  halves  of  the       ^ 
curve  we  must  break  up 


40 


DEVELOPMENT    IN    TRIGONOMETillC    SERIES.  [Art.  26. 


j /(.t)  sin  itix.dx     into      j  f{x)  sin  mx.dx  +    |  /(a-)  sin  mx.dx. 

<i  0  n- 

2 

We  have,  then, 

«,„  =    I  -i'  sin  mx.dx  +    |  (tt  —  x)  sin  mx.dx 


(1) 


=  -^  sill  //^ 


But                     sin  ni  —  =  1       if       m  ^=  1       or  4,k  +  1 

=  0        ''       m  =  2        "  4A'  +  2 

=  -  1  ''       w  =  3        "  4A-  +  3 

=  0        ''        m.  =  4        "  4A: . 
Hence  if     y  =  /(,r)     represents  our  broken  line  , 

„,  ,         4  /sin  .T       sin  3a-    ,    sin  oic  sin  Ix    ,          \ 


When     .,=2     /(.r)=| 


and  Ave  have 


8        1-       3^       5-       7- 
(rf)     As  a  case  where  the  function  has  a  linite  discontinuity,  let 

f(a-)  =  1     from     x  =  0     to     -t  =  —       and 

f(x)  :=  0         "         •'^  =  17     "       .r  =  TT  . 

//  ^  f(.r)     will  in  this  case  represent  the  locus  in  the  figure. 
T  As  before 


//(x) 


(2) 


(3) 


sin  mx.dx  =   I  f(x)  sin  mx.dx 


sin  mx.dx  . 


in 


2  r  ■  -  C 

=  —    I  sin  mx.dx  -\ 10. 


sin  mx.dx . 


(1) 


Chap.  II.]  EXAMPLES,                                                               41 

2    f  .  ^21/.                   7r\ 

a,,,  =  —    f  sm  nix.dx  = (1  —  cos  m  —  I  • 

TT  ^  IT  in  \                              If 
TT 

But                  cos  m  -  =      0  if       ?«-  =  1       or       4/.-  +  1 

=  -  1  "       m  =  2        "        4/c  +  2 

=      0  "       m.  =  3        "        4A-  +  3 

=      1  ''       w^  =  4        "        4A; 

Hence 


,    ,         2  /sin  ic    ,   2  sin  2.r    ,   sin  3j'    ,   sin  5a'    ,   2  sin  6,r    ,   sin  7a:;  ,        \ 

TT  1 

If     X  =  -     the  second  member  of  (2)  reduces  to  -  ,  for 

and  we  see  that  the  series  represents  the  function  completely  for  all  values  of 
X  between  x  =  0  and  x  =^  —  except  for  x  =  —  and  there  it  has  a 
value  which  is  the  mean  of  the  values  approached  by  the  function  as  x 
approaches  7^  from  opposite  sides. 

EXAMPLES. 

Obtain  the  following  developments :  — 

,,,       2        2  r/TT^      4\    .  TT^    .     _      ,    /TT^       4\    .     _  -rr'    .      . 

(1)  ^  \   \~\ Ts I  s^^  *'  —  ':^  sm  2.r  +  I  — I  sin  dx  —  —  sm  4a? 

2    r/TT^  ()7r\      .  /TT^  67r\  „         ,      /TT^  ()7r\  .-, 

(2)  »■'  =  i  L(t  -  p)  ^™  ^'-  -\2-l?)  '"'  -■'■  +  (3  -  3?)  ^'"  •^»' 

/o^      /./  N        2  rsina-    ,    tt    .     _  sin  3a;       27r    .     ,      ,    sinSoJ 

(3)  /(a-)  =  -  1^^^  +  -,  sm2a;  -  -^  -  ^  sm  4a;  +  -^ 


.    37r    .      _  -| 

+  -^  sm  6a'  —  •  •  •  J  , 


42  DEVELOPMENT    IX   TEIGOXOMETEIC    SERIES.  [Art.  27 

TT  77 

if  /(x)  =  X   from  .r  =  0  to  ■»'  ^  -^  and  /(.r)  =  0  from  .r  ==  —   to  ./■  =  tt. 

.,       .  2    .  r   sin.r  2sin2.r    ,    3  sin  .'l/'       4sin4./'    ,  ~1 

(4)  sm  fix  =  -  sm  fiir      —, :,  —  ~, :,  +  —, :,  — :,  +  "  "  • 

if  /i  is  a  fraction. 

2  ri  2  .s 

(5)  e-f  =  —     -  (1  +  e'")  sin  .'r  +  r  (1  —  e"")  sin  2.r  +  tt:  (1  +  ^'^)  sin  '^^ 

+  ^_  (1  -  '-)  sin  4x  +  •  •  •]  . 

/^N       ■   1  2  sinliTT  rl.  2.^,3._  4.,.        -I 

(6)  smh  ,T  =  -  sm  a-  —  r  sm  Jx  -\-  —■  sm  3.r  —  t^  sm  4./-  -\ 

^  ^  7T  \_-  o  10  It  J 

2  ri  .2 

(7)  cosh  a*  =  —      -  (1  +  cosli  tt)  sin  ,/■  +  ~  (1  —  cosli  tt)  sin  2x 


+  ;^  (1  +  cosh  tt)  sin  3.r  H ~| 


10 


27.     Let  us  now  try  to  develop  a  given  function  of  aj  in  a  series  of  cosines. 

As  before  suppose  that  f(x)  has  a  single  value  for  each  value  of  x  between 
a-  =  0  and  x  ^=  tt,  that  it  does  not  become  infinite  between  x  =  0  and 
X  :=  TT,    and  that  if  discontinuous  it  has  only  finite  discontinuities. 

Assume 

f(x)  =  bo  +  bi  cos  X  -\-  ho  cos  2x  -\-  h^  cos  3.''  +  ■  •  ■  (1) 

To  determine  any  coefficient  6,„  multiply  (1)  by  cos  vix.dx  and  integrate 
each  term  from  0  to  tt. 

j  Z-o  cos  mx.dx  ^  0. 

TT  TT 

j  h,.  cos  kx  cos  mx.dx  =  ^ "j  [cos  (m  —  //).'-  +  t'os  (in  +  k)x']dx 

0  (I 

=  0  if  m  and  k  are  not  equal. 

h„,  cos^  mx.dx  "^  -^  fz/ix  +  cos  ma;  sin  mx), 
2m  ^ 

/>„,  cos'^  mx.d.v  =  —  &„j ,  if  7n  is  not  zero. 

2    /•  2    /^ 

Hence  Z>„,  =  —   j  f(x)  cos  mx.dx  =  —   j  /(a)  cos  ma.da  ,  (2) 

n  0 

if  m  is  not  sero. 


Chap.  II.]  EXAMPLES    OF    COSINE    SEIllES.  43 

To  get  Ji^y  multiply  (1)  by  d.r  and  integrate  from  zero  to  tt. 

I  ^^.  cos  kxjJ.r  =  0. 
Hence  h,=   -   Cf(x)dx  =  - f{a)da,  (3) 

TT  J  TT 

which  is  just  half  the  value  that  would  be  given  Ijy  formula  (2)  if  zero  were 
substituted  for  m. 

To  save  a  separate  formula  (1)  is  usually  written 

f(x)  =  i^o  ~t~  ^^1  cos  X  -\-  1)2,  cos  2x  +  ^3  cos  3,r  +  •  •  •  (4) 

and  tlien  the  formula 

2    r  2   /i  . 

J^  =  —    I  f(x)  cos  mx.dx  =  —    j  f{a)  cos  via.da  (2) 

0  0 

will  give  (iq  as  well  as  the  other  coefficients. 

It  is  important  to  see  clearly  that  what  we  have  just  done  in  deter- 
mining the  coefficients  of  (1)  is  equivalent  to  taking  n  -\-  1  terms  of  (4), 
substituting  in 

1/  =  4-/^5  +  I'l  cos  X  +  h.  cos  2,r  +  •  •  •  +  ^„t  cos  nx  (5) 

in  turn  the  coordinates  of  the  n  +  1  points  of  the  curve 

whose  projections  on  the  axis  of  A' are  equidistant,  determining  h,  bi,  b^,  •  •  •  /*„ 
by  elimination  from  the  n  +  1  resulting  equations,  and  then  taking  the  limit- 
ing values  they  approach  as  n  is  indefinitely  increased,      (v.  Art.  24.) 

TT 

If  Ax  ^=  — q— :  the  abscissas  of  the  n  +  1  points  used  are  0,  Ax,  2 Ax-, 
3A.r ,  •  •  •  uAx,  so  that  we  should  expect  our  cosine  development  to  hold  for 
x  ^  0     as  well  as  for  values  of  x  between  zero  and  tt. 

28.     Let  us  take  one  or  two  examples  : 

(a)     Let  /'(.t)  =  X .  (1) 

TT  J  TT   J 

2    r  2  2 

J„  =  -   1  X  COS  mx.dx  —  --r  (cos  vitt  —  1 )  =  -^  [(—  1)  '"  — 11. 


"i-i  DEVELOPMENT    IN    TIUGONOMETRIC    SERIES.  [Art.  28. 


Hence      .r 


■^        -1  /  cos  o.r    ,    COS  5.r    ,    cos  7.1 


■^         *i  /  I    cos  o.r       cos  t).r    ,    cos  i  x    ,  \ 

(2)  holds  good  not  only  for  values  of  x  between  zero  and  tt  but  for     x  =  0 
and     X  =  TT    as  well,  since  for  these  values  we  have  - 

which  are  true  by  Art.  26  ('.')(3). 

(b)     Let  f(x)  =  .7-  sin  .r  .  (1) 

2    r  2 

^U  =  -     I  ./•  sin  X.l/X  =  -  TT  =:  2 , 
TT  J  TT 


and  __-,    4/.    ,    1    ,    1    ,    1 


*■  =  - 

TT 


j  ./■  sin  a-  cos  x.f/x  =  -   r,/'  sin  2x.dx  =  —  -  , 
r.r  sin  .r  cos  mx.dx  =  -   C [_x  sin  {in  +  l).r  —  x  sin  (///  —  l)^](Za; 


2 


if  ^y;  is  odd 


{ni  -  l){m  +  1) 

2 

^  — z — r^rr     if  III  is  even. 

(/;/  —  !)(/»  +  1) 

Hence 

.     .        cos  X       2  cos  2x    ,    2  cos  3.r    .  2  cos  4.'"    , 

X  sin  ,/■  =  1  —      — \-  •  '  •  (T\ 

2  1.3      ^      2.4  3.5      ^  ^"'' 


Jf     ^'  =  o     we  have 


l=\  +  h-h  +  h---  (^) 


EXAMPLES. 
Obtain  the  following  developments: 


(1)      f{^) 


_  TT  _  2  Fcos  2x       cos  6a?       cos  lOx       cos  14.r  n 

~  4       TT  L     P      "^      3^     "^  ^5^~  +  ~T^        ^  ■  ■  ■  J 


if     f{x)  =  a;  from  cf  =  0  to  a-  =  -  and  /(a;)  =  tt  —  .r  from  a-  =  ^  to  ,r  =  tt  . 


Chap.   II.]  EXAMPLES.  45 

(2)   /(.)  =  1  +  2  piip  _  ^:^  -+  '^  _  '^  +  ^  .-| . 

^  TT  \_       L  O  O  i  _J 

if     f(x)  =  1   from  .r  =  (»  to  .r  =  -  and  ,/'(./■)  =  ()  from  x  =  ^  to  ./■  =:  tt. 

,„.        „       TT^        .  rcos  ,/•       cos  U.r    ,    cos  o,i'       cos  4.r    ,  n 

(3)  a-  =  --4|_^^ ^  +  -S' 4^+-J- 

(4)  a'^  =.  —  ---  |^(^—  -  pj  cos  .r  -  ^  cos  2x  +  ^^  -  .Tij  cos  3x 

-  ^  cos  4,r  +(?;;-  ^^,)  cos  5.r J  . 

(5)  /(.r)  =  I  +  I  [^^^  -  l)  cos  ,r  -  I  cos  2,r  -  |  ( y  +  l)  cos  3.r 

+  -^  / -7; 1 )  cos  5,z-  —  -^^  cos  6j-  —  •  •     , 


TT 


if       f{x)  =  ,r    from    .r  =:  0    to    .r  =  -  and  /(.r)  =  0    from    :r  ^  -;^    t( 

2  ri 


)  cos  2./; 


(6)    «'  =  ^  [2  (<"  - 1 )  -  iTT'  (<•■  + 1)  ™^  ■'■  +  r^,  ("-  -  1 : 

,^,  ,  2  sinh  TT  rl        1  ,    1  o  1 

(/)     cosh  .j^-  =  o  —  o  cos  a?  +  -  cos  2.r  —  —-  cos  ox 

TT  l_J  J  O  10 

+  -p.  cos  4,T  —  ■  ■     • 

2  rl  1 

(8)  sinh  a;  =  —     -  (cosh  tt  —  1)  —  -  (cosh  tt  +  1 )  cos  x 

+  r  (cosh  TT  —  1)  cos  2,r  —  T—  (COsh  IT  -\-  I)  cos  o.r  +  •  ■  •      . 

.„.  2/x  sin  /LtTT  r  1  cos  .r      ,     cos  2x  cos  .S.r 

(9)  cos  ^lx  =  -^= ^     — ., ^ +  - ., .-, - 

7r  \_2/x-       jx-  —  1^       fi-  —  2"       /u-  —  .V 

cos  4.r  ~1 

+  /I^^^T^  -  •  •  •  J  ' 

if  /i  is  a  fraction. 

29.  Although  any  function  can  be  expressed  both  as  a  sine  series  and  as  a 
cosine  series,  and  the  function  and  either  series  will  be  equal  for  all  values  of 
X  between  zero  and  tt,  there  is  a  decided  difference  in  the  two  series  for  other 
values  of  x. 

Both  series  are  periodic  functions  of  ,/■  liaving  the  period  27r.  If  then  we 
let  //  equal  the  series  in  question  and  construct  the  portion  of  the  correspond- 


40 


DEVELOPMENT    IN    TRIGONOMETKIC    SERIES. 


[Art.  29. 


iug  curve  Avhich  lies  between  the  values     x  =  —  tt     and     x  =  it     the  whole 
curve  will  consist  of  repetitions  of  this  portion. 

Since  sin  mx  =  —  sin  (—  mx)  the  ordinate  corresponding  to  any  value  of 
X  between  —  tt  and  zero  in  the  sine  curve  will  be  the  negative  of  the  ordinate 
corresponding  to  the  same  value  of  x  with  the  positive  sign.  In  other  words 
the  curve 

y  =  cii  sin  X  -\-  a^  sin  2x  -\-  «3  sin  3x  -{-  •  •  •  (1) 

is  symmetrical  with  respect  to  the  origin. 

Since  cos  mx  =  cos  ( —  mx)  the  ordinate  corresponding  to  any  value  of  x 
between  —  tt  and  zero  in  the  cosine  curve  will  be  the  same  as  the  ordinate 
belonging  to  the  corresponding  positive  value  of  x .     In  other  words  the  curve 

1/  =  ^bo  +  bi  cos  X  -\-  b2  cos  2x  +  b^  cos  3x  -\-  •  •  •  (2) 

is  symmetrical  with  respect  to  the  axis  of  Y. 

If  then  f(x)  =^  —  /( —  x) ,  that  is  if  /(a?)  is  an  odd  function  the  sine  series 
corresponding  to  it  will  be  equal  to  it  for  all  values  of  x  between  —  tt  and  tt, 
except  perhaps  for  the  value  x  =  0  for  which  the  series  will  necessarily  be 
zero. 

If  f(x)  =f(—x),  that  is  if  f(x)  is  an  even  function  the  cosine  series  cor- 
responding to  it  will  be  equal  to  it  for  all  values  of  x  between  x  =  —  tt  and 
X  =  TT,    not  excepting  the  value     a'  =  0. 

As  an  example  of  the  difference  between  the  sine  and  cosine  developments 
of  the  same  function  let  us  take  the  series  for.  x . 


sin  2x    ,    sin  ox 


y  =  2  [sii 

TT  4  r  ,      cos  3.r      ,     COS  O.C 

//  =  t;  —  -      COS  X  H -, h 


?in  4. 


+ 


3^       ■       5^ 

[v.  Art.  26(rt)  and  Art.  2S(«)].      (3)  represents  the  curve 


(3) 
(4) 


/ 

and  (4)  the  curve 


Chap.  II.]  FOUMEK's    SERIES.  47 

Both  coincide  with  y  =  x  from  x  =  0  to  x  =  tt,  (3)  coincides  witli 
y  =z  X  from  x  =  —  ir  to  .i;  =  tt  ,  and  neither  coincides  with  y  z=.  x  for 
values  of  x  less  than  —  tt  or  greater  than  tt.  Moreover  (3),  in  addition  to 
the  continuous  portions  of  the  locus  represented  in  the  figure,  gives  the  iso- 
lated points  (—  7r,0)  (7r,0)  (37r,0)  &c. 

30.  We  have  seen  that  if  f(x)  is  an  odd  function  its  development  in  silfe 
series  holds  for  all  values  of  x  from  —  tt  to  tt  ,  as  does  the  development  of 
f(x)  in  cosine  series  if  f{x)  is  an  even  function. 

Thus  the  developments  of  Art.  26(^0>  ^i"t-  26  Exs.  '(2),  (4),  (6);  Art.  28 (^<) 
Art.  28  Exs.  (3),  (7),  (9)  are  valid  for  all  values  of  x  between  —  ir  and  tt. 

Any  function  of  x  can  be  developed  into  a  Trigonometric  series  to  which  it 
is  equal  for  all  values  of  x  between  —  tt  and  tt. 

Let  fix)  be  the  given  function  of  x.  It  can  be  expressed  as  the  sum  of  an 
even  function  of  x  and  an  odd  function  of  x  by  the  following  device. 

identically ;   but    '-^^-^ — -^ — '—     is  not  changed  by  reversing  the  sign  of  x  and 

is   therefore  an    even    function    of   x\    and  when  we    reverse  the  sign  of  x, 

"-^^ — ~ is  aifected  only  to  the  extent  of  having  its  sign  reversed  and 

is  consequently  an  odd  function  of  x . 

Therefore  for  all  values  of  x  between  —  ir  and  tt 

/(■^■)  -\-f{-  ■^)  ^  1  ^^^  _^  ^^^  ^^g  ^,  ^  ^^  ^Qg  2,r  +  /,3  cos  3.r  +  •  •  • 
where  h,,^  =  -  j  ^-^^^ — ^ — --'  cos  mx.dx  ;  and 

u 

-'^^ f^ —  =  (I  I  sin  X  -\-  (u  sin  2x  -\-  a^  sin  3.r  +  •  •  • 

2   n  f(  ^')  —  f( —  x) 
where  a...  =  -  |  '^^^ — -f^ — ^  sin  mx.dx . 

-rrj  2 

^,„  and  o„j  can  be  simplified  a  little. 

t,.  =  Jf/W +/(-''■)  eos«xA 
ttJ  2 

=  -       I  /(a-)  cos  mx.dx  -\-    i  f{—  x)  cos  mx.dx     , 


48  DEVELOPMENT   IN    TRIGONOMETRIC    SERIES.  [Art.  30. 

but  if  we  replace  x  by  —  x,  we  get  , 

77  —IT  0 

I  /'( —  x)  COS  mx.dx  =  —   I  f(x)  cos  mx.dx  =   |  /(.r)  cos  mx.dx , 

and  we  have  h„^  =  —  |  /(»•)  cos  mx.dx . 

In  the  same  way  we  can  reduce  the  vahie  of  a,^  to 
—   I  j\x)  sin  mx.dx  . 


Hence 


f(x)  =  -  bo  +  &i  cos  ^  +  ^2  COS  2x  +  ^.^s  cos  3cc  + 


(2) 


where 


and 


+  a  I  sin  X  +  "2  sin  2x  +  f^3  sin  3x  +  • 
6,„  =  —  j  f(x)  cos  mx.dx  =  —   j  /(a)  cos  ma. da.  (3) 

1   /^  .  1   /^  . 

a,„  ^  —  I  /(cr)  sin  mx.dx  =  —  j  /(a)  sin  ma.da  .  (4) 


and  this  development  holds  for  all  values  of  x  between  —  tt  and  tt.  ^ 

The  second  member  of  (2)  is  known  as  a  Fourier's  Series. 

EXAMPLES. 

1.   (,)btain  the  following  developments,  all  of  which  are  valid  from  x  =  —  tt 
to  j^'  =  7r  :  — 


(1)      e-^  = ^^-  _     cos  a^  +  p  cos  2,r-  —  —  cos  8x  +  —  cos  4.r  H 

7r| -,        Z  o  lU  li  I 

,    2sinh7rrl    .  2    .     ,,      ,    3     .     .,  \  -| 

H sm  X  —  -  sm  2x  -\-  — -  sm  ox  —  — ;  sin  4.7^  +  •  •  •     . 

TT  |_J  O  10  1<  _| 

TT      2  r  cos  3.7-       COS  5.r       cos  7x  ~\ 

sin  ,r       sin  2x       sin  3.r       sin  4.r 
"^^^  2~+~3  r~+'"' 

where  /(ic)  =  0  from  .r  =  —  tt  to  x  =  0  and  /(.r)  =  x  from  j'  =  0  to  x  =  tt. 


Chap.  II.]  EXTENSION    aF    FOURIEIl's    SERIES.  49 

(3)    /(^)  ~  "  T?  +  ~     T-2  cos  a;  +  -  cos  2x  +  -  cos  3x  +  -  cos  5x 
+  ^^  cos  Qx  +  •  •  •  J 


-TT  Sin 


where    f(x)  =  x  from  a;  —  —  tt  to  .^-  =  0 ,  f(x)  =  0  from  x  =  0  to  .i?  =  -^  > 

TT  TT 

and        /(.r)  =  a'  —  —     from    .r  =  ^^    to    x  =  tt  . 

2.  Show  that  formula  (2)  Art.  30  can  be  written 

.%)  =  9«o  COS  /3o  +  ('i  COS  (X  ~  /8i)  +  r,  COS  (2.r  —  (3.2)  +  6-3  COS  (3x  —  iSg)  H 

where  (•„  =  («„;  +  />„-;) "'     a^iid     /3„j  =  tan"  ^  ~  ■ 

3.  Show  that  formula  (2)  Art.  30  can  be  written 

/(a)  =  1 6-0  Sin  ^0  +  r,  sin  (,/■  +  ^0  +  r,  sin  (2x  +  (3,)  +  ^-3  sin  (3.^  +  P,)  +  ■  ■  • 
where  c,„  =  (a,-  +  bjj^     and     I3„,  =  tan~^  -^  • 

31.  In  developing  a  function  of  x  into  a  Trigonometric  series  it  is  often 
inconvenient  to  be  held  within  the  narrow  boundaries  .r  =  —  tt  and  x  =  tt  . 
Let  us  see  if  we  cannot  widen  them. 

Let  it  be  required  to  develop  a  function  of  x  into  a  Trigonometric  series 
which  shall  be  equal  to  /(.z-)  for  all  values  of  x  between  x  =  —  c  and  x  =  c . 

Introduce  a  new  variable 

TT 
Z  =:  —  X  , 

which  is  equal  to  —  tt  when  x  =  —  c  and  to  tt  when  x  =  c. 
/(.r)  =,//-  -)     can  be  developed  in  terms  of  z  by  Art.  30  (2),  (3),  and  (4). 
We  have 


f{z  ')  =  9  ^^"  +  ^  ^os  z  +  b,  cos  2z  +  b,  cos  3,-  +  •  •  •  )  ^^^ 

-\-  «i  sin  z  +  (u  sin  2,?  +  ffg  sin  3z  -{-■■■  ) 

where  6„,  =  -    j /(-  ■s )  cos  /msj.c?^;  .  (2) 


50  DEVELOPMENT    IN    TKIGONOMETRIC    SERIES.  [Art.   31. 

and  ^™  ~  ^  f'^TT    /  ^^^^  ^nz.dz  .  (3) 

and  (1)  holds  good  from     s  =  —  tt     to     z  ^  ir  . 
Replace  z  by  its  value  in  terms  of  x  and  (1)  becomes 

/..  N       1  7     1    7          ''''•^17           27ra-    .    -          Sttx    . 
f(x)  =  -hQ-\-b^  cos  —  +  Ih  cos  — h  h  cos  — 1-  •  •  • 

.     7r.r    ,            .      2irx    ,            .      .STT.r    ,  '  ^^ 

+  «i  sm  ■ 1-  a.2  sm f-  a^  sm +  • 

c  c  c 

The  coefficients   in  (4)   are  the   same  as   in  (1),  and  (4)   holds   good   from 
X  =  —  (■     to     X  =  c  . 

Formulas  (2)  and  (3)  can  be  put  into  more  convenient  shape. 


or  5,„  =  -    \  f(^)  cos  —  f^a:;  =  -  j  /(A)  cos c?A  .  (5) 

—  c  — c 

In  like  manner  we  can  transform  (3)  into 

«m  =  -  J  /(a-)  sm  -^  dx  =  -J  f(X)  sm  -^  rfA  .  (6) 

By  treating  in  like  fashion  formulas  (1)  and  (2)  Art.  25  and  formulas  (4) 
and  (2)  Art.  27  we  get 

^_  ,  .     TTX    ,  .     2'Trx    .  .     Sttx    .  ^„^ 

f(x)  =  a,  sm \-  (u  sm 1-  cu  sm \-  '  '  '  (1) 

^  c  c  c  ^ 

where  ««  —  "7    1  /(^')  sin  dx=-i  /(A)  sin  — ^  rZA .  (8) 

tl  0 

and     f(x)  :=  -  ho -{-  hi  cos  —  +  ^»2  cos 1-  b^  cos  — ^ r  •  '  •       (9) 

where     ^»„,  =  -  I  f(x)  cos dx  =  -    i   /(A)  cos dX  .  (10) 

0  0 

and  (7)  and  (9)  hold  good  from    .r  =  0    to    x  =  c . 


Chap.  II.]  EXTENSION    OF    FOURIER's    SERIES.  51 

EXAMPLES. 
1.    Obtain  the  folloAving  developments: 

(1)  1  =-•  -     sm \-  -  sm +  p  sin h  "  •  ' 

from    .T  ^  0    to    .r  =  c  . 

2c  r  .     7r,r       1    .     2'jr.T    ,    1     .     Stt.t        1     .     47r.r    ,  "1 

(2)  X  ■=  —     sm sm 1-  -  sm sin h  *  "  * 

^^  IT  \_         r         2  c  6  c  4:  c  J 

from    a-  ^  —  c    to    x  =.  e . 

c        4''  r        7r.r    ,     1  37ra;    ,    1  Bttx        1  Ttt./-    ,  "I 

.T  = ^,     cos \-  -,  cos h  —  cos 1"  -  cos h  •  ■  • 

2       IT-  \_         e         3^  c  D^  c  (-  c  J 


from    X  =  0    to    x  =  c 


„,        ,       2r-^r/7r2        4\     .     TTX        7r2    .     2'7rx    ,     /tt^        4\     .     37r.r 

(3)  ^■■'  =  ^LVT  -  1^)  ^^^^  T  -  Y  ^^^^  -T  +  (t  -  ,^^)  ^^"  — 

TT-      .       iTTX      ,      (IT-  4\       .        7)17  X      .  ~\ 

__3ni  — +  (^--;sm-^+-.-J 

from    X  =  0    to    j:  =:  c . 
fA       4^-2  p        ^^,        -^  27rx    ,    1  Sttx        1         47r./; 

•T^  =  TT ~  \    COS cos h  02  COS  —  y^  COS  

3        TT-  L         '■         -^  c         3^  c         4^  c 

from    ./•  =  —  r    to    X  =  (' . 

r.      ri  +  t'"  7r.r    ,    2(1—^^)     .     27rx 

(4)  e^  =  27r       .,7"     -2  s"^ ^     )   ,     ,    .;  sm  

,    3  (1  +  e'-)    .     Sttx    ,    A(l  —  ef)    .     Attx    ,  "] 

+     0    1    a    -   s^" ^    o\    -, ..    o  sm h  •  •  •     > 

c-  -\-  97r-  c  t-  +  IbTT-  ('  J 

^       _     ri  e/  - 1         e'^  +  1  7r.x    ,      ^'  -  1  2'rrx 

e^  =  2c\  -  — — cos h    .2   ,    .    .,  cos 

L2      c^  c^  +  TT^  c         c^  +  47r''  c 


cos  '-^  +  •  •  •  J 
from   X  =  0    to   ',r  =  c  . 


c"  -\-  1  oTT.r 


from   .'P  =  0   to   X  =  c , 

c  c 

where    f(x)  =  a;   from   x  =  0   to   a;  =  -   and   f(x)  =  c  —  x   from   .a^  =  :^    to 


52  DEVELOPMENT    IN    TRIGONOMETRIC    SERIES.  [Akt.  ■^2. 

2.  Show  that  formula  (4)  Art.  .'31  can  be  written 

f(x)  =  -  r,  COS  ^0  +  ''1  COS   {^—  -  (3,)  +  Co  COS  (^-—  -  (3o) 

4-  03  COS  /^  —  M  +  ■  •  • 

where  ^-^  =(«,'  + ^„l)^     and     ^„,  =  tan-i^. 

3.  Show  that  formula  (4)  Art.  .31  can  be  written 

1  /TT.r  \  /27r.r  \ 

/(«')  =  r,  '0  sin  ^0  +  '-1  sin  (^—  +  (3^)  +  'v  sm  (-^  +  /i^j 

+  .3  sin  (•^"  +  a)  +  •  • 

where  r-„,  =  («J  +  ^>,f)*     and     /3„  =  tan-^  ^  . 


32.  In  the  formulas  of  Art.  31  c  may  have  as  great  a  value  as  we  please, 
so  that  we  can  obtain  a  Trigonometric  Series  for  f(x)  that  will  represent  the 
given  function  through  as  great  an  interval  as  we  may  choose  to  take.  If, 
then,  we  can  obtain  the  limiting  form  approached  by  the  series  (4)  Art.  31  as 
c  is  indefinitely  increased  the  expression  in  question  ought  to  be  equal  to  the 
given  function  of  x  for  all  values  of  x.  Equation  (4)  Art.  31  can  be  written 
as  follows  if  we  replace  ^o?  ^u  ^25  "  *  '  Ui,  a^,  •  •  •  by  their  values  given  in 
Art.  31  (o)  and  (6). 

-H  I  /(A)  cos  -y  cos  —  ''A  +  I  /(A)  cos  — —  cos c/A  +  •  •  • 

+  Cf(X)  sin  —  sin  —  r/A  +  Cf(X)  sin  ^^^  sin  ^^^  (ZA  +  •  •  •  | 


1    /* ,    N    7.    rl    ,  ''■A  TT.l-    .      .     ttA     .     TT.r 

T    I  /(^)  ^^^  \  V  ~'~  *-**^^  ~r  *^*"^  ~T  "^  '^^^^  ~T  ^^^^  ~T 


27rA         27ra"    ,      .     27rA    .     27r.i 
-j-  cos  — —  COS \-  sm  sin  


Chap.  II.]  FOURIER'S   INTEGRAL.  53 

/■(.r)  -  i  ff(X)dX   [^  +  cos  ^  (A  -  .r)  +  cos  ^  (A  -  .>■)  +  •  •  -1 

=  ^_  ff(K)dX   fl  +  cos  ^  (A  -  .'■)  +  cos  y  (A  -  .'■)  +  •  •  • 

+  cos  (-  ^)  (A  -  *•)  +  cos  (-  =^)  (A  -  ./■)  +  •  •  -1 
since  cos  ( —  <^)  =  cos  <^  . 

.m = ^  ffi^yi^  [•  •  • + 7  «o«  (-  v)  ^^  - "') + 7  •^'^^  (-  7)  (^  -''^^ 

—  c 

H cos  -^  (A  —  .r)  H —  COS  —  (A  —  X) 

+  _  COS  —  (A  -  ./■)  +  •  •  •  J  (1) 

As  c  is  indefinitely  increased  the  limiting  value  approached  by  the 
parenthesis  in  (1)  is 

I  cos  a(A  —  x).(la. 
Hence  the  limiting  form  approached  by  (1)  is 

f(x)  =  ^  J f(\)d\y cos  a  (A  -  .r).da  ,  (2) 

and  the  second  member  of  (2)  mnst  be  equal  to  f(^x)  for  all  values  of  x . 

The  double  integral  in  (2)  is  known  as  Fou/'ier's  Integral,  and  since  it  is  a 
limiting  form  of  Fourier's  Series  it  is  subject  to  the  same  limitations  as  the 
series. 

That  is,  in  order  that  (2)  should  be  true  f(x)  must  be  finite,  continuous,  and 
single  valued  for  all  values  of  x,  or  if  discontinuous,  must  have  only  finite 
discontinuities.* 

(2)  is  sometimes  given  in  a  slightly  different  form. 
X  0  3: 

Since       |  cos  a  (A  —  x).da  =  |  cos  a  (A  —  x).da  -\-  j  cos  a  (A  —  x).da 

and 

0  II  (I 

I  cos  a(\  —  x).da  =    j  cos  (—  a) (A  —  x).d(—  a)  =  —    |  cos  a(A  —  x).da 

i  cos  a(A  —  x).da  =  2  |  cos  a  (A  —  x).da 
*  See  note  on  pajxe  P>S. 


54  DEVELOPMENT    IN    TRIGONOMETRIC    SERIES. 

and  (2)  may  be  written 

/(.r)  =  -    Cf(\)dX  fcos  a(A  —  x).da  .  (3) 

If  f(x)  is  an  even  function  or  an  odd  function  (3)  can  be  still  further  simpli- 
fied. 

Let  /(.^.)  =-/(-.,;). 

Since  the  limits  of  integration  in  (3)  do  not  contain  a  or  X  the  integrations 
may  be  performed  in  whichever  order  we  choose.     That  is 

ff(X)dX  fcos  a(X  —  x^.da  =  Cda  Cf(X)  cos  a(X  —  a:).dX  . 

—  o=  0  0  —=c 

Now 
Cf(X)  cos  a(X  —  x).dX  =  Cf{X)  cos  a{X  —  x).dX  +  Cf(X)  cos  a(X  —  x).dX  . 

Cf(X)  cos  a(X  —  x).dX  =  Cf{—  X)  cos  a(—  X  —  x).d{—  X) 
=  —   Cf{X)  cos  a{X  +  x).dX 

0 

and  (3)  becomes 

f(x)  =  -  Cda  Cf(X)  [cos  a(X  —  x)  —  cos  a(X  +  x)^.dX 

=  —  (da  I /(A)  sin  aX  sin  ax.dX 

or  f(x)  =  -  Cf(X)dX  I  sin  aX  sin  ax.da  .  (4) 

If    f(x)  =  /( —  x)     (3)  can  be  reduced  in  like  manner  to        1    -  /-^ 

f(x)  =  -   Cf(X)dX  j  cos  aX  cos  aa^.rfa  .  (5) 

Although  (4)  holds  for  all  values  of  x  only  in  case  f(x)  is  an  odd  function, 
and  (5)  only  in  case  f(x)  is  an  even  function,  both  (4)  and  (5)  hold  for  all 
positive  values  of  x  in  the  case  of  any* function. 

EXAMPLE. 

(1)     Obtain  formulas  (4)  and  (o)  directly  from  (7)  and  (9)  Art.  31. 


CHAPTER    TIL 

COXVERGENCE    OF    FOURIER's    SERIES. 

33.  The  question  of  the  convergejire  of  a  Fourier's  Series  is  altogether  too 
large  to  be  completely  handled  in  an  elementary  treatise.  We  will,  however, 
consider  at  some  length  one  of  the  most  important  of  the  series  we  have 
obtained,  namely 

4  r  .  ,    sin  3.r    ,    sin  hx    ,    sin  7.r    ,  ~\ 

-  I  sm  X  +  --g \ ^ \ h  •  •  •     .     [v.  (3)  Art.  L^6(//).] 

and  prove  that  for  all  values  of  x  between  zero  and  it  its  sum  is  absolutely 
equal  to  unity;  that  is,  that  the  limit  approached  by  the  sum  of  n  terms  of  the 
series 

-      sin  X  I  sin  a.da  +  sin  2x  j  sin  la.da  +  sin  ox  Ain  oa.da  +  •  •  •      , 

as  n  is  indefinitely  increased,  is  1,  provided  that  x.  lies  between  zero  and  tt. 
Let 

'^w  =  ~     sin  X  I  sin  a.da  +  sin  'Ix  |  sin  la.da  +  sin  '.\x  |  sin  oa.da  +  •  •  • 

+  sin  nx  j  sin  na.da     .  (1) 

Then 
S„=  ~    I  [sin  a  sin  x  -\-  sin  2a  sin  2x  -\-  sin  3a  sin  ox  +  •  •  •  +  sin  /i.a  sin  nx'\<hi 

0 

1   r 

=  —   I  [cos  (a  —  .r)  —  cos  (a  +  x)  +  cos  2(a  —  ,/•)  —  cos  2(a  +  a-)+  •  •  ■ 

+  cos  n(a  —  x)  —  cos  n(a  +  x)^da 

1  Jl 

=  -  I  [cos  (a  —  X)  +  cos  2 (a  —  .r)  +'Cos  3(a  —  ,r)  +  •  ■  •  +  cos  ii(a  —  x)yia 

j  [cos  (a  +  x)  -\-  cos  2(a  +  a-)  +  cos  3(a  +  .r)  +  •  •  •  +  cos  )i{a  +  .>")]^^«  • 


56  CONVERGENCE    OF    FOUEIER'S    SERIES.  [Art.  84. 

Therefore  by  Art.  20  (1) 

'rrJ\_J       1  .     a  —  X  J 

0  sm  — - — 

u  sm  ^— 

^     ^sin(2.  +  l;^  ^     ^sin(2.  +  l)^ 

'^"^2^j  — -7-^^r — ^^"-2^J  — r^+:r— ^"' 

()  sm  — - —  0  sm  — - — 

In  the  first  integral  substitute  ^  for    — — ^ ,    and  in  the  second  integral  sub- 
stitute /3  for     — -y^  . 
We  get 

ttJ  sm  B  ^      ttJ  sm  B  ^  ^ 


It  remains  to  find  the  limit  approached  by  /S^  as  n  is  indefinitely  increased. 

34.  E 

pin(2„  +  l)/i       ^^.  (1) 

./  sm  yS  2 


Fo 


^''Vsin"^  ^^^  =  i  +  cos  2/3  +  cos  4/3  +  ■  •  •  +  cos  2?z^ ,  by  Art.  20. 

and  rcos2A-/3.(//3  =  0. 

Let  us  construct  the  curve 

sin  (2??  +  l).r 

1/  = ^-^ —  . 

sm  X 

We  have  only  to  draw  the  curve  //  =  sin  (2ti  +  l)x  and  then  to  divide 
the  length  of  each  ordinate  by  the  value  of  the  sine  of  the  corresponding 
abscissa. 

In  1/  =1  sin  (2n  -\-  l)x  the  successive  arches  into  which  the  curve  is 
divided  by  the  axis  of  A' are  equal,  and  consequently  their  areas  are  equal. 


SU31MATI0X    OF    A    SINE    SEKIES. 


57 


Each  arch  ]ms  for  its  altitude  unity  and  for  its  base 
and  is  symmetrical  witli  respect  to  the  ordi- 


2.7,  +  1  _ 

nate  of  its  highest  or  lowest  point. 


sin  (2».  +  l).r 
sin  X 


■  If  now  we  form  the  curve    //  ^ ^-^ — ' — '—     from 

the  curve     i/  =  sin  (2ii  -\-  l)x  ,    it  is  clear  that,  since 

TT 

sm  X  increases  as  x  increases  from  0  to  —  ,  the  ordi- 
nate of  any  point  of  the  new  curve  Avill  be  shorter 
than  the  ordinate  of  the  corresponding  point  in  the 
preceding   arch,  and   that    consequently  the  area  of 

each  arch  of     //  =  ' — '-     will    be   less    than 

sin  x 

that  of  the  arch  before  it. 

If  fto?  «ij  ^h,  '  ■  ■  '^n-i  ai'6  the  areas  of  the  suc- 
cessive arches  and  ((i^  that  of  the  incomplete  arch  termi- 
nated by  the  ordinate  corresponding  to     x  =  — 


sin  (2n  +  l)x 


dx  —  r/y  —  (i^  +  ^'2  —  "3  +  •  •  •  • 


in  {2n  +  l).x-  ,  r  sin  {2n,  +  1)^  tt 


/ 


sin  /? 


dli  =  i,  by(l). 


<i^  —  "\  +  "'z  —  ''3  +  "4  ~  ■  ■  ■  +  <in    if  '^  is  even, 

^  =  (1^,  — (^  -(-  (In  —  II.  +  ^'4  —  •  •  —  Hn     ii^  "•  is  odd. 
These  equations  can  be  written 

^      ZT^ ,,,  +  (-  ''1  + ''.;)  +  (-  <>.  +  ''4) 

if  a  is  even,  and 

5  =  "0  +  (- "1  +  "2)  +  (- "3  + ''4) 

+  (-  "5  +  «c)  +  •••  +  (-"«-.  +  <-'n  -  1)  +  (-  ''«) 

if  n  is  odd. 


68  rONVERGENCE    OF    FOUEIER's    SERIES.  [Art.  35. 

In  either  case  each  parenthesis  is  a  negative  quantity  since 

tto  >  «i  >  «2  ^  "3  ^  (f'nf 

and  it  folloAvs  that  r/o  is  greater  than  —  • 
Again 

TT 

-  =  ((0  —  a  I  +  (r/2  —  rts)  +  (a4  —  r/5)  -i [-  (r/„  _2  —  «„  _i)  +  % 

if  ??  is  even  and 

TT 

-  =  r,„  —  ,,j  -f  (r/2  —  r/3)  +  («4  —  r/5)  +  •  •  •  +  (rt„  _j  —  aj 

if  7i  is  odd. 

In  either  case  each  parenthesis  is  j)Ositive  and  it  follows   that    a^  —  «i   is 

less  than   — . 


Since 


tto  and  (Iq  —  (I I  differ  from    —    by  less  than  they  differ  from  each  other,  that 
is,  by  less  than  a^ . 

In  like  manner  we  can   show  that    «o  —  ^1    and    a,-,  —  c/i  +  "^s    differ   from 

TT 

—  by  less  than  a.y,  and  in  general  that    r/,,  —  "1  + '/o  — <^3  +  '  '  "  i '^t    differs 

■^  TT 

from    -  by  less  than  fij.;  or  even  that 

^'0  —  ''1  +  «2  —  f'3  +   ■    •    •  ±  -' 

differs  from  —  by  less  than  r/^.  no  matter  what  the  value   of  p ,  provided  p  is 
greater  than  unity. 

35.     Prom  what  has  been  proved  in  the  last  article  it  follows  that 

sin  (2)1  -\-  1)3 


f^""~-        * 


sm  X 

TT  ,     TT 


where  b  is  some  value  between   7; — r— -    and  —  ,    differs  from   —    by  less  than 

2/^  +  1  2 '  2      "^ 

sill  (^)i  "4"  l^x 

the  area  of  the  arch  in  which  the  ordinate  of     u  = ''"^- —      correspond- 

sm  X 

ing  to   X  =  I)    falls  if  this  ordinate   divides  an  arch,  or  by  less  than  the  area 

of  the  arch  next  beyond  the  point  (b ,  0)  if  the  curve  crosses  the  axis  of  A'  at 

that  point. 


Chap.  III.]  SUMMATION    OF    A    SINE   SERIES.  59 


TT 


The  area  of  the  arch  in  question  is  less  than    - — — — - ,  its  base,  miTltiplied  by 
1 


a  value  greater  than  the  length  of  its  longest  ordinate. 


h 

r  sin  (2«  +  l)x  , 

iheretore  I   ^-^ ^  aa 

J  sm  X 

differs  from    —  by  less  than    ,:.^ — —:, 
2     -^  III  +  1 


sm  lb—  - — —  I 
V        2n  +  1/ 

TT  1 

If   now   n  is  indefinitely  increased    - — -— approaches 

^  1,1+1        .       /  TT        \ 

zero  as  its  limit,  and  we  get  the  very  important  result 

h 

limit   r  r  sin  (2ii  -\-  l)x      n  _  tt  ,^^ 

^=  2o  |_J  sin  ic  "^  J  ~  2 


if  ()<h<'^ 


ttJ  sin/3  ^       ttJ  sm;8  '^  ^  ^ 

"i  2 

_1    /^  Sin  (2.  +  1)/?  1  Y sin  (2n.  +  l)/3 

~7rJ  sin^  ''^  +  7rJ  sin /3  '^^ 

IT  X 

_1    A  Sin  (2^.+  !)^  1   fsin_(2«  +  l)_^ 

ttJ  sin^  '^J  '"'^ 

lYsin(2».+  l)^ 

This  last  value  for  »S'„  can  be  somewhat  simplified. 
Substituting     y  =  —  /3     we  get 

X 

r  sin  (2n  +  1)^       ^       r  sin  (2;^.  +  l)y  ^^    _  pin  (2.^  +  l)/3  ^^  ^ 
J  sinfi  J  sinv  ^     J  sinfi 


60  CONVERGEXCE    OF    FOUEIEll'S    tiElUES.  [AuT.  37. 

Substituting     y  =  tt  —  /3     in 

n-       X 

2+2 

/sin  (2n  +  V)R  ,^  , 
!^ ! — '"  dR                                       we  have 
sin  /3 


psin(2n  +  l)^  /-sin(2y^  +  l)y^^    ^  p  sin  (2m  +  l)/3  ^^ 

J  sin  /iJ         '  J  sin  7  ^      J  sin  /^ 


0  0 

Hence 

_  2  pin(2M  +  l)/3  2Ysm(2M  +  l)^  2  pin  (2»  +  l)/3 

^^"-ttJ  sin^  '^J  sin/3         '^'^      ttJ  sin^         '^^• 

pin(2.,  +  l)/3  TT  ^^.^_  3^_ 

J  sin/3  '^2  "^   ^ 

0  ' 

and 

linntrpm(2.  +  l)/3       n       TT      .^     o  <  ,.  <  vr       by  (1)  Art.  35. 

Therefore  limit   pon_i  +  i_i=,i     if     0<.r<7r  and 

4  r  .         ,   sin  3.T   ,   sin  S.r   .   sin  Ix   .         "1       . 

for  all  values  of  x  between  zero  and  tt. 

37.  By  a  somewhat  long  but  not  especially  difficult  extension  of  the  rea- 
soning just  given  it  can  be  shown  that  if  f{x)  is  single-valued  and  finite 
JQetween  x^  —  tt  and  x  =  vr,  and  has  only  a  fimfe  mnvher  of  discon- 
tinuities and  of  maxima  and  minima  between  x'  =  —  tt  and  x  =  vr  the 
Fourier''s  Sevies 

o  ^''o  +  l>\  COS  x  +  hn  cos  2x  +  />3  cos  3.T  +  •  •  ■ 
2 

+  f/j  sin  X  +  (^'o  sin  2,r  +  «3  sin  3a"  -j-  •  •  • 


Chap.  HI.]  DIKICHLET's   CONDITIONS.  61 

TT 

where  a^-=-  \  f{a)  siu  ma.da 

and  &,„  =:  -  j  f(a)  cos  ma.da  , 

and  that  Fourier's  Series  only  is  equal  to  f(x)  for  all  values  of  x  between 
a;  =  —  TT  and  x  =  ir ,  excepting  the  values  of  x  corresponding  to  the  discon- 
tinuities of  f(x),  and  the  values  tt  and  —  tt  if  /(tt)  is  not  equal  to  /(—  tt); 
and  that  if  c  is  a  value  of  x  corresponding  to  a  discontinuity  of  f(x),  the  value 
of  the  series  when    x  =^  c    is 

1  limit    ^  ,,^         ,    ,     .      .     . -, 

and  that  if  /(tt)  is  not  equal  to  /(—  tt)  the  value  of  the  series  when  x  =  —  ir 
and  when    a-  =:  tt  is  ' 

If  f(x)  while  satisfying  the  conditions  named  in  the  preceding  paragraph 
except  for  a  finite  number  of  values  of  x,  becomes  infinite  for  those  values,  the 
series  is  equal  to  the  function  except  for  the  values  of  x  in  question  provided 

that        ('fi-'c)dx      is  finite  and  determinate,      (v.  Int.  Cal.  Arts.  83  and  84.) 

38.  The  question  of  tlie  convergency  of  a  Fourier's  Series  and  the  condi- 
tions under  which  a  function  may  be  developed  in  such  a  series  was  first 
attacked  successfully  by  Dirichlet  in  1829,  and  his  conclusions  have  been 
criticised  and  extended  by  later  mathematicians,  notably  by  Riemann,  Heine, 
Lipschitz,  and  du  Bois  Reymond.  It  may  be  noted  that  the  criticisms  relate 
not  to  the  sufficiency  but  to  the  necessity  of  Dirichlet's  conditions. 

An  excellent  resume  of  the  literature  of  the  subject  is  given  by  Arnold 
Sachse  in  a  short  dissertation  published  by  Gauthier-Villars,  Paris,  1880, 
entitled  "Essai  Historique  sur  la  Representation  d'une  Fonction  Arbitraire 
d'une  seule  variable  par  una  Serie  Trigonometrique." 

89.  A  good  deal  of  light  is  thrown  on  the  peculiarities  of  trigonometric 
series  by  the  attempt  to  construct  approximately  the  curves  corresponding  to 
them. 

If  we  construct  y  =  a^  sin  x  and  y  =  a.2  sin  2x  and  add  the  ordinates 
of  the  points  having  the  same  abscissas  we  shall  obtain  points  on  the  curve 


t)2  CONVEKGENCE    OF    FOUKIEK'S    SERIES.  [Art.  39. 

ij  =  c/j  sin  X  -\-  a^  sin  2x . 

If  now  we  construct  y  =  (h  sin  ?>x  and  add  the  ordinates  to  those  of 
//  r=  11^  sin  X  -\-  «2  sin  2x     avb  shall  get  the  curve 

//  =  «!  sin  ./•  -|-  do  sin  2.x  +  a^  sin  3.t. 

By  continuing  this  process  we  get  successive  approximations  to 

//  =  a-^  sin  X  +  no  sin  2a:'  +  r/3  sin  ox  -\-  a^  sin  \x  4-  •  •  • 

Let  us  apply  this   method   to  a  few  of  the  series  which  we   have  obtained  in 
Chapter  II. 
Take 

?/  ^  sin  a?  +  -  sin  '?,x  +  -  sin  5.t  +  •  •  •  (1) 

IT 

=  0    when    x  ^  0 ,    -    from   x  =  0  to  .1;  =  tt  ,  and  0  when  x^ir , 
4 

V.  Art.  26  [/>](3). 

//  =  2  ^sin  X  —  \  sin  2,r  +  ~  sin  3,/-  —  J  sin  4.r  +  •  •  ■)  (2) 

=  ./■    from    .r  =  0    to    .r=:7r,    and   0   when    .r  =  7r, 
Art.  26[a.](4). 

//  =  -     r-o  sin  X  —  -  sin  3.r  +  p^  sin  o.r  —  -,_  sin  7.7-  +  •  •  •  (3) 

=  .r    from    x  =  0    to    a-  =  —  ,    and    ir  —  x    from   x  =  -   to   .r  =  tt  . 
Art.  2G  [r](2). 

12  112  1 

y  =  -  sin  X  +  -  sin  2x  -\-  -  sin  3.'"  +  -  sin  5./ —  -  sin  6x  +  -  sin  7x-\-  ■■  ■  (4) 

1  w  o  o  O  ( 

TT  TT  TT 

^0  when  .c  =  0,  —  from  .r  =  0  to  .'•  =  —  ,  and 0  from  '''  =  -^  to  x  =  7r. 
v.  Art.  26  [./](2). 

It  must  be  borne  in  mind  that  each  of  these  curves  is  periodic  having  the 
period  27r,  and  is  symmetrical  with  respect  to  the  origin. 

The  following  figures  I,  II,  III,  and  IV  represent  the  first  four  approxima- 
tions to  each  of  these  curves. 

In  each  figure  the  curve  i/  =  the  series,  and  the  approximation  in  question 
are  drawn  in  continuous  lines,  and  the  preceding  approximation  and  the  curve 
corresponding  to  the  term  to  be  added  are  drawn  in  dotted  lines. 


Chai'.  111.]      SUCCE 


^CESSIVE    APPilOXlMATlOlS'S    TO    A    SINE    SEKIES.  63 


Y 

0/ \y 

"7  ^  "v" 


Y 

/■ — ^<' "~>^^ — 

"\ 

/  y 

-A 

0 

/•'                          "^N                                    / 

-^\ 

X 

/ 

V.^'' 

TC 

\ 

Y 

7\-^.     — .c— jc^ 

''7" 

0 

,-'"~^,            .'-'"N 

-> 

X 

t 

V.y           'v../' 

"• '' 

71 

I 

(54 


CONVEllGENCE    OF    FOUKIER's    SERIES.  [Art.  .39. 


Chap.  TII.l  PROPERTIES    OF    FOURIER  S    SERIES.  6o 

Figs.  I,  II,  III,  and  IV  immediately  suggest  the  following  facts : 

(a)  The  curve  representing  each  approximation  is  continuous  even  when 
the  curve  representing  the  series  is  discontinuous. 

(b)  When  the  curve  representing  the  series  is  discontinuous  the  portion  of 
each  successive  approximate  curve  in  the  neighborhood  of  the  point  whose 
abscissa  is  a  value  of  x  for  which  the  series  curve  is  discontinuous  approaches 
more  and  more  nearly  a  straight  line  perpendicular  to  the  axis  of  X  and  con- 
necting the  separate  portions  of  the  series  curve. 

((•)  The  curves  representing  successive  approximations  do  not  necessarily 
tend  to  lose  their  wavy  character,  since  each  is  obtained  from  the  preceding 
one  by  superposing  upon  it  a  wave  line  whose  waves  are  shorter  each  time  but 
do  not  necessarily  lose  their  sharpness  of  pitch.  This  is  the  case  in  Figures 
I,  II,  and  IV.  In  Fig.  Ill  the  waves  of  the  superposed  curves  grow  rapidly 
flatter. 

It  folloAvs  from  this  that  in  such  cases  as  those  represented  in  Figures  I,  II, 
and  IV  the  direction  of  the  approximate  curve  at  a  point  having  a  given 
abscissa  does  not  in  general  approach  the  direction  of  the  series  curve  at  the 
corresponding  point,  or  indeed,  approach  any  limiting  value,  as  the  approxima- 
tion is  made  closer  and  closer;  and  that  the  lengj;h  of  any  portion  of  the 
approximate  curve  will  not  in  general  approach  the  length  of  the  correspond- 
ing portion  of  the  series  curve. 

Analytically  this  amounts  to  saying  that  the  derivative  of  a  function  of  x 
cannot  in  general  be  obtained  by  differentiating  term  by  term  the  Fourier's 
Series  which  represents  the  function. 

(d)  The  area  bounded  by  a  given  ordinate,  the  approximate  curve,  the  axis  of 
X,  and  any  second  ordinate  will  approach  as  its  limit  the  corresponding  area  of 
the  series  curve  if  the  series  curve  is  continuous  between  the  ordinates  in 
question;  and  will  approach  the  area  bounded  by  the  given  ordinate,  the  series 
curve,  the  axis  of  X,  any  second  ordinate,  and  a  line  perpendicular  to  the  axis 
of  A',  and  joining  the  separate  portions  of  the  series  curve  if  the  latter  has  a 
discontinuity  between  the  ordinates  in  question. 

Analytically  this  amounts  to  saying  that  the  Fourier's  Series  corresponding 
to  any  given  function  can  be  integrated  term  by  term  and  the  resulting  series 
will  represent  the  integral  of  the  function  even  when  the  function  is 
discontinuous  (v.  Int.  Cal.  Art.  83). 

We  may  note  in  passing  that  if  the  function  curve  is  continuous  a  curve 
representing  the  integral  of  the  function  will  be  continuous  and  will  not 
change  its  direction  abruptly  at  any  point;  while  if  the  function  curve  is  dis- 
continuous the  curve  representing  the  integral  will  still  be  continuous  but  will 
change  its  direction  abruptly  at  points  corresponding  to  the  discontinuities  of 
the  aiven  function. 


66  CONVERGENCE    OF    FOUKIER's    SEKIES.  [Art.  40. 

40.  The  facts  that  the  derivative  of  a  Fourier's  Series  cannot  in  general  be 
obtained  by  differentiating  the  series  term  by  term  and  that  its  integral  can  be 
obtained  by  integrating  the  series  term  by  term  are  so  important  that  it  is 
worth  while  to  look  at  the  matter  a  little  more  closely.  Let  us  consider  the 
differentiation  of  the  series  represented  in  Art.  39  Figure  I. 

Let 

Sn  =  sin  a-  +  -  sin  3^"  +  -  sin  ox  +  •  •  •  +  ^ — j—r  sin  (2n  +  l)ic . 

Then  ~  =  cos  x  +  cos  3a-  +  cos  5x  -\-  ■  ■  •  -{-  cos  (2m  +  l)x. 

ax 


u   .  =  1 


f"=0 

ax 


and  the  curve  is  parallel   to  the  axis  of  X  for    x  =  -     no  matter  what  the 

value  of  ii . 

If    a.-  =  0    or    X  =  IT 

^"  =  1  +  1  +  1  +  1  +  •  •  ■  +  1  =  v/  +  1 

ax 
and  the  curve     //  =  S^     becomes   more  nearly  perpendicular  to  the  axis  of  X 
at  the  origin  and  for    x  =z  tt    as  we  increase  n. 

If  .  =  1 

dx       2  '^22  2 

That  is  — r^'  =      -     if     n  =  0     or     n  =  ^k 

dx 

=  —  -     ••      /i  =  1      "      n  =  3/t  +  1 

=      0     ''      w  =  2      -'      n  =  Sk  +  2. 

Consequently  when    x  =  —      ~'  does  not  approach  any  limiting  value  as  n  is 

indefinitely  increased.      Indeed,  in  the    successive  approximations  the  point 

whose  abscissa  is  —  is  successively  on  the  rear,  on  the  front,  and  on  the  crest 
o 

or  in  the  trough  of  a  wave,  and  although  the  waves  are  getting  smaller  they  do 

not  lose  their  sharpness  of  pitch. 

If  X  has  any  other  value  between  0  and  tt     — "  will  change  abruptly  as  n  is 
changed  and  will  not  approach  any  limiting  value  as  n  is  increased. 


Chap.  III.]  DIFFERENTIATION    OF    FOURIEIi's    SERIES.  67 


41.  In  general  if  we  differentiate  a  Fourier's  Series 

'S'  =  9  <^o  +  ^'\  COS  :i:  +  ho,  cos  2.r  +  !>.  cos  ox  +  •  •  • 

+  c/j  sin  .r  +  (U2  sin  2,r  +  r/3  sin  o^  +  '  *  * 
we  get 

—  hi  sin  X  —  2Z'2  sin  2x.  —  Sh^  sin  3.«  —  •  •  • 

+  <(i  cos  x  +  2a2  cos  2.i'  +  3a s  cos  o.«  +  •  •  • . 
Differentiate  again  and  we  get 

—  hi  cos  x  —  2'\  cos  2x  —  3%s  cos  3,> —  •  •  • 

—  «!  sin  X  —  2^(f2  sin  2.t —  oVj^  sin  3j'  —  •  •  ■ . 

We  see  that  each  time  we  differentiate  we  multiply  the  coefficient  of  sin  7cx 
and  of  cos  kx  by  k  while  the  term  still  involves  cos  kx  or  sin  kx . 
Since  the  series 

cos  X  +  COS  2x  -\-  cos  ox  +  •  •  • 
+  sin  X  +  sin  2x  +  sin  3x  +  •  •  • 

is  not  convergent,  and  a  Fourier's  Series  converges  only  because  its  coefficients 
decrease  as  we  advance  in  the  series,  the  differentiation  of  a  Fourier's  Series 
must  make  its  convergence  less  rapid  if  it  does  not  actually  destroy  it,  and 
repetitions  of  the  process  will  usually  eventually  make  the  derived  series 
diverge. 

It  is  to  be  observed  that  the  derived  series  are  Fourier's  Series,  but  of  some- 
what special  form,  that  is  they  lack  the  constant  term.     (v.  Art.  30.) 

If  now  we  integrate  a  Fourier's  Series 

-  h„  -f  hi  cos  X  +  ho  cos  2x  +  hs  cos  3x  +  •  •  • 
+  a  I  sin  X  -\-  a^  sin  2x  +  ag  sin  3.^  +  •  •  • 
we  get  ^  "^  9  ^■'o'^"  ^~  ''^i  ^^^^  ^  ~^  ~  ^'"  ^^^^  ^^'  +  ~  ^3  ^^^^  '^•■*^  -\-  ■  " 

1  O  1  Q 

—  r/i  cos  a —  -  «2  cos  2x  —  7-  a^  cos  ox  —  •  •  • , 

2  o 

a  Trigonometric  Series  which  converges  more  rapidly  than  the  given  series. 

It  is  to  be  observed  that  the  series  obtained  by  integrating  a  Fourier's 
Series  is  not  in  general  a  Fourier's  Series  owing  to  the  presence  of  the  term 
^box.     (V.  Art.  30.) 

42.  We  are  now  ready  to  consider  the  conditions  under  which  a  function  of 
X  can  be  developed  into  a  Fourier's  Series  whose  term  by  term  derivative  shall 
be  equal  to  the  derivative  of  the  function. 


68  CONVERGENCE    OF    FOURIEU'S    SERIES. 

Let  the  function  f(x)  satisfy  the  conditions  stated  in  Art.  37.  Then  there 
is  one  Fourier's  Series  and  but  one  which  is  equal  to  it.     Call  this  series  *S'. 

Let  the  derivative  /'(«)*  of  the  given  function  also  satisfy  the  conditions 
stated  in  Art.  37.  Then  f'{x)  can  be  expressed  as  a  Fourier's  Series.  By  Art. 
39  (d)  the  integral  of  this  latter  series  will  be  equal  to  the  integral  of  f\x)^ 
that  is  to  f(x)  plus  a  constant,  and  one  integral  will  be  equal  to  f(x) . 

If  this  integral  which  is  necessarily  a  Trigonometric  Series  is  a  Fourier's 
Series  it  must  be  identical  with  S.  It  will  be  a  Fourier's  Series  only  iu  case 
the  Fourier's  Series  for  /'(.r)  lacks  the  constant  term  ^h^. 


But 


^jy'(,x)dx  by  (3)  Art.  30. 


Therefore  h^  =  -  [/(tt)  —  /{—  tt) ] ; 

and  will  be  zero  if    /\7r)  =/'(—  tt). 

In  order  that  f'(x)  shall  satisfy  the  conditions  stated  in  Art.  37  f(x)  while 
satisfying  the  same  conditions  must  in  addition  be  finite  and  continuous 
between    x  =  —  tt    and    cc  =  tt. 

If,  then,  /(.r)  is  single-valued,  finite,  and  continuous,  and  has  only  a  finite 
number  of  maxima  and  minima,  het^eew  .r  =  —  tt  and  .r  =  tt,  (the  values 
a:  =  —  TT  and  x  =^  ir  being  included),  and  if  /(tt)  =/( — tt)  f(x)  can  be 
developed  into  a  Fourier's  Series  whose  term  by  term  derivative  will  be  equal 
to  the  derivative  of  the  function. 

It  will  be  observed  tKat  in  this  case  the  periodic  curve  y  =  S  is  continuous 
throughout  its  whole  extent. 

43.  Since  a  Fourier's  Integral  is  a  limiting  case  of  a  Fourier's  Series  the 
conclusions  stated  in  this  chapter  hold,  mutatis  mtitandis  for  a  Fourier's 
Integral. 

For  example  if  a  function  of  x  is  finite  and  single-valued  for  all  values  of  x 
and  has  not  an  infinite  number  of  discontinuities  or  of  maxima  and  minima  in 
the  neighborhood  of  any  value  of  x  it  Avill  be  equal  to  the  Fourier's  Integral 


Ifiajm 


COS  a  (A  —  x).d\ 

and  to  that  Fourier's  Integral  only,  and  the  integral  with  respect  to  x  of  this 
Fourier's  Integral  will  be  equal  to      Cf(x)dx. 

If  in  addition  /(.r)  is  finite  and  continuous  for  all  values  of  x  the  derivative 
of  the  Fourier's  Integral  with  respect  to  x  will  be  equal  to      ^y7  • 

*  We  shall  regularly  use  the  notation  f'{x)   for   '—j^  .     v.  Dif.  Cal.  Art.  124. 


CHAPTER   IV. 

SOLUTION      OF     PROBLEMS     IN      PHYSICS      BY     THE     AID     OF     FOURIER'S 
INTEGRALS    AND    FOURIER's    SERIES. 

44.  In  Art.  7  we  have  already  considered  at  some  length  a  problem  in 
Heat  Conduction  which  required  the  use  of  a  Fourier's  Series.  We  shall  begin 
the  present  chapter  with  a  problem  closely  analogous  in  its  treatment  to  that 
of  Art.  7,  but  calling  for  the  use  of  a  Fourier's  Integral. 

Suppose  that  electricity  is  flowing  in  a  thin  plane  sheet  of  infinite  extent 
and  that  the  value  of  the  potential  function  is  given  for  every  point  in  some 
straight  line  in  the  sheet,  required  the  value  of  the  potential  function  at  any 
point  of  the  sheet. 

Let  us  take  the  line  as  the  axis  of  X  and  consider  at  first  only  those  points 
for  which  y  is  positive: 

We  have,  then,  to  satisfy  the  equation 

BIV^DIV=^  (1) 

subject  to  the  conditions 

F=0       when     >j=^cr>  (2) 

F=/(.x-)      "        ^  =  0  (3) 

where  f{x)  is  a  given  function,  and  we  are  not  concerned  with  negative 
values  of  //. 

As  in  Art.  f  we  have  e~""  sin  ax  and  e~"^  cos  ax  as  particular  values  of  F 
which  satisfy  (1)  and  (2).  We  must  multiply  them  by  constant  coefficients 
and  so  combine  them  as  to  satisfy  condition  (3). 

By  (3)  Art.  32 

f(x)  =  -  Cda  Cf(X)  cos  a  (A  —  x)  .dX .  A) 

We  wish  to  build  up  a  value  of  F which  will  reduce  to  (4)  when  //  =  (). 
This  requires  a  little  care  but  not  much  ingenuity. 


70  SOLUTION    OF    PROBLEMS    IN    PHYSICS.  [Airr.  4i. 

Take  e^^^cosaa.-  and  e~'^^sinax  and  multiply  the  first  by  cos  aA,  and 
the  second  hj  sin  aA ;  they  are  still  values  of  V  which  satisfy  (1).  Add 
these  and  we  get 

e-«^cosa(A  — .t)j 

still  a  value  of  V  which  satisfies  (1),  no  matter  what  the  values  of  a  and  A. 
Multiply  by  f(\)dX   and  we  have 

e~''yf(X)  cos  a(A  —  x).dk  (5) 

as  a  value  of  V  which  satisfies  (1). 

V=  Ce-'^i' f(\)  cos  a(\  —  .r).dX  (6) 

is  still  a  solution  of  (1)  since  it  is  the  limit  of  the  sum  of  terms  covered  by 
the  form  (5);  and  finally 

V=  -  Cda  Ce- <'yf{\)  cos  a(A  —  x).dX  (7) 

is  a  solution  of  (1)  as  it  is  —  multiplied  by  the  limit  of  the  sum  of  terms 

TT 

formed  by  multiplying  the  second  member  of  (6)  by  da  and  giving  different 
values  to  a. 

But  (7)  must  be  our  required  solution  since  while  it  satisfies  (1)  and  (2),  it 
reduces  to  (4)  when   ?/^0    and  therefore  satisfies  condition  (3). 

If  f(x)  is  an  even  function  we  can  reduce  (7)  to  the  form 

F=  -  j  da  r^""^/(A)  cos  ax  cos  aA.c^A  (8) 

and  if  f(x)  is  an  odd  function  to  the  form 

2  r      r 
r=-  I  da  I  e-'^yfik)  sin  ax  sin  aX.dX.  (9) 

(7),  (8),  and  (9)  are  valid  only  for  positive  values  of  ?/,  but  as  the  problem  is 
obviously  symmetrical  with  respect  to  the  axis  of  A'.  (7),  (8),  and  (9)  enable 
us  to  get  the  value  of  the  potential  function  at  any  point  of  the  plane. 


EXAMPLES. 

1.  Obtain  forms  (8)  and  (9)  directly  by  the  aid  of  (5)  and  (4)  Art.  32. 

2.  State  a  problem  in  statical  electricity  of  which  the  solution  given  in 
Art.  44  is  the  solution. 


Chap.    IV.]       FLOW    OF    ELECTRICITY    IN    AN    INFINITE    PLANE.  71 

45.  As  a  special  case  under  Art.  44  let  us  consider  the  problem :  —  To  find 
the  value  of  the  potential  function  at  any  point  of  a  thin  plane  sheet  of  infinite 
extent  where  all  points  of  a  given  line  which  lie  to  the  left  of  the  origin  are 
kept  at  potential  zero,  and  all  points  which  lie  to  the  right  of  the  origin  are 
kept  at  potential  unity. 

Here   f(oc)  =  0   if   a-<0    and   /(.r)  =  1    if   x>0. 

(7)  Art.  44  gives  us  the  required  solution.     It  is 

T'= -  Cda  Ce- -^^ cos  a(A  —  .iS).dX ;  (1) 

but  this  can  be  much  simplified. 
We  have 

F=-  j  flk  I  e~"2'cosa(A  —  x).da. 

Now  i  e~"^  cos  vix.dx  =^  -^—, -, 

J  a-  -\-  »r 

0 

if    ^/  >0.     (Int.  Cal.  Art.  82,  Ex.  8.) 
Hence 


/ir'^y  cos  a  (A  —  x).da  =  -^ — ^ 
=  1  C y^ =  -  (''^  +  tan-  ^  '^Y 


tan  I  —  —  tan-^  - )  =  ctn  ( tan"  ^  - )  ^  - : 
V2  y/  \  y/       x' 

and  consequently 

F=-(?  +  tan-2:)  =  l_itan-^.  (2) 

TT  \2  I//  TT  X  ^ 

Since  log  z  =  log  (.«  +  ijl)  =^  -  log  (x^  +  //-)  +  *  tan-  ^  - , 

[Int.  Cal.  Art.  33  (2)], 

;  -  -  log  ^  =  /•  -  -  log  (X  +  yi)  =  -  —  log  (x'  +  y-')  +  /  (l  -  -  tan" '  ^) 


If     -1.'/           1  1 

—  tan    ^  -     and — 

Cal.  Arts.  209  and  210.)     Hence 


and    1 tan    ^  -     and     —  -^  log  (x-  +  y'^)     are  conjugate  functions,     (v.  Int. 


ri  =  -^log(.r2  +  2/2)  (3) 

is  a  solution  of  the  equation 

DlV,ArD-^V,  =  ^',  (4) 


72 


SOLUTION    OF    PKOBLEMS    IN    PHYSICS. 


[Akt.  45. 


and  the  curves 


-  (-  + tan-'  -)  =  a 


(5) 


and 


-  —  log(.r'  +  ,f)=b 


(6) 


cut  each  other  at  right  angles. 

If  we  construct  the  curves  obtained  by  giving  different  values  to  a  in  (5)  we 
get  a  set  of  equipotential  lines  for  the  conducting  sheet  described  at  the  begin- 
ning of  this  article,  and  the  curves  obtained  by  giving  different  values  to  (h)  in 
(6)  will  be  the  lines  offloiv. 


Moreover  since 


Vi=  — 


—  log  {x^  +  y/2) 


(3) 


is  a  solution  of  Laplace's  Equation  (4),  the  lines  of  flow  just  mentioned  will  be 
equipotential  lines  for  a  certain  distribution  of  jiotential,  for  which   the  equi- 
"potential  lines  above  mentioned  will  be  lines  of  flow. 
F=a,     that  is 


^(J  +  tan-'^) 


(5) 

reduces  to  U'^~  ^  ^^^^  '^'^  ■  (^) 

If  now  we  give  to  a  values   differing  by  a  constant  amount  we  get  a  set  of 
straight  lines  radiating  from  the  origin  and  at  equal  angular  intervals. 
V-^  =  b,     that  is 


-iTT 


b, 


(6) 


reduces  to 

x^-^,/^e-'^\  (8) 

If  we  give  to  i  a  set  of  values  differing  by  a  constant  amount  we  get  a  set 
of  circles  whose  centres  are  at  the  origin  and  whose  radii  form  a  geometrical 
progression.  They  are  the  equipotential  lines  for  a  thin  plane  sheet  of  infinite 
extent  where  the  potential  function  is  kept  equal  to  given  different  constant 
values  on  the  circumferences  of  tAvo  given  concentric  circles  or  where  we  have 

a  source  as  the    origin;    and    for  this 
i  system  the  lines  (7)  are  lines  of  flow, 

and  (3)  is  the  complete  solution. 

The  figure  gives  the  equipotential 
lines  and  lines  of  flow  for  either  sys- 
tem, but  only  for  positive  values  of  y. 
The  complete  figure  has  the  axis  of  X 
as  an  axis  of  symmetry. 


Chap!    IV.]       FLOW    OF    ELECTRICITY    IN    AN    INFINITE    PLANE.  73 

EXAMPLES. 
1.    Solve  the  problem  of  Art.  44  for  the  case  where 

/(.x)=  — 1     if     x<0     and    /(■*■)  =  1     if     .t>0. 


2  :r 

A71S.,      V^  -  tau"^ ^  - . 

TT  // 


2.    Solve  the  problem  of  Art.  44  for  the  case  where 


f(x)  =  a     if     X  <  0     and    f(x)  =  h     if     .;•  >  0 . 

Ans.,     V=  \  (<i  -\-b)  +  -(b  —  a)  tan" ^  -  . 

'   3.    Reduce  (7),  (8),  and  (9)  Art.  44  to  the  forms 
Try.  ^'+ (A -a;)''' 

respectively. 

46.     An  especially  interesting  case  of  Art.  44  is  the  following  where 
f(x)  =  0     if    x<-l,    f(x)  =1    if    -  1<  .r  <  1  ,    and    f(x)  =0    if    x>l. 

V=-  ftan- 1  ^^^^  +  tan- '  ^^^^  .  (1) 


Here 


Kow         -  log  [(1  -  z)ir\  =  -  log  [(1  -  X  -  ///)/]  =  -  log  [//+  (1  - ./■)*] 


l-log[(l-..)^  +  /]  +  ^tan-^. 


and 


-  ;^  log  [(-  1  -  ;^) /]  =  -  _^  log  [(-!-./•-  yO  C  =  -  ^  log  [y  -  (1+  x)q 
^-~  log  [(1  +  .ry  +  //]  +  ^ tan- '  ^^' . 
-  log  — =  ~-  log  Vn ,      •    +  -     tan-  ^ h  tan   ^ . 


74  SOLUTION    OF    PEOBLEMS    IN    PHYSICS.  [Art.  4(j. 

Hence 

vrV  //  //    /  'Jtt     °  (1 -\- X)- -\- i/' 

are  covjuf/ate  functions;  *  and 


-  (tan- 1  ^-i£  +  tan- '  -^ -)  =  «  (2) 

is  any  equipotential  line,  and 

1      ,  (1  -  Xf  +  /  ,  ,ON 

any  line  of  flow  for  the  system  described  at  the  beginning  of  this  article;  and 

is  the  solution  of  a  new  problem  for  which  (3)  represents  any  equipotential 
line  and  (2)  any  line  of  flow. 


The  function  conjugate  to 
1 


tan-i h  tan-i 

\_  y  'J  J 


miglit  have  been  found  as  follo\v.s.     If  4>  is  the  required  function  and  \p  the  given  function  we 
have  by  Int.  Cal.  Arts.  208,  209,  and  210  the  relations 

Dx(p  =  D,f\f/     and     D,j4>  =  —  D^-^p . 

If  now  we  integrate  Dyf  with  respect  to  x  treating  ?/  as  a  constant  and  add  an  arbitrary 
function  of  y  we  shall  have  0 .     So  that 

.^  =  -  :;^  { log  [(1  +  x)-^  +  y^]  -  log  [(1  -  xf  +  y']  {  +Ay) . 


_       1  r  y y  ~|   I  '^ 

^•""^  ~       TT  |_  ( 1  +  x)-^  +  7/      (1  -  x)--^  +  7/-^  J  "^  " 


1%) 


Comparing  this  with  its  equal   —  D^V'   above  we  find     -^r^  =  0    and    f(y)  —  C    a  constant 

therefore  ^  log  ^^'""tl'^'^  +  ^ ' 

27r     ''  (1  +  X)-  +  ?/- 

where  C  may  be  taken  at  pleasure,  is  our  required  conjugate  function. 


CiiAP.  IV.]  SOURCE    AiSD    SINK   IN    AN    INFINITE   PLANE. 

(2)  reduces  to 

——. — '— ==  tan  (ITT 

x'  +  y'  —  i 

or  x^  +  (//  —  etn  (nry~  =  csc-«7r  ; 


and  (3)  to 


^■^+//^+2 


e^'^+l 


+  1  =  0 


75 


(5) 


(.r  +  ctnh  ^vtt)-^  +  y-  =  csq.Ii'-Ik 


(6) 


(5)  and  (6)  are  circles.  The  circles  (5)  have  their  centres  in  the  axis  of  Y, 
and  pass  through  the  points  ( —  1 ,  0)  and  (1 ,  0) ;  and  the  circles  (6)  have  their 
centres  in  the  axis  of  A'. 

(4)  is  the  complete  solution,  (6)  is  any  equipotential  line  and  (o)  any  line  of 
flow  for  a  plane  sheet  in  which  the  points  in  the  circumferences  of  two  given 
circles  whose  centres  are  further  apart  than  the  sum  of  their  radii  are  kept  at 
different  constant  potentials,  or  where  a  source  and  a  sink  of  equal  intensity 
are  placed  at  the  points  ( — 1,0)  and  (1, 0).  An  important  practical  ex- 
ample is  where  two  wires  connected  with  the  poles  of  a  battery  are  placed 
with  their  free  ends  in  contact  with  a  thin  plane  sheet  of  conducting  material. 
The  figure  shows  the  equipotential  lines  and  lines  of  floAV  of  either  system. 

The  complete  figure  would  have  the  axis  of  A'  for  an  axis  of  symmetry. 


v=o  If^„    v=i     v,-^         v=o 

EXAMPLES. 


1.    Show  that  if    /(.;•)  =  (?i    when    :f<i  —  h,    f(.v)  =  a2   Avhen    —h<^x<.b, 
f(x)  =  rt3    Avhen    x  >  b  , 


r=^^+^ 


-\^(a,-a,) 


-ith 


tan-^  — ' 1-  (c/2  —  ''3)  tan-^ - 


76  SOLUTION    OF    riiOBLEMS    IX    PHYSICS.  [Akt.  47. 

2.    Show  that  if   f(x)=()    if   ,v<0,  f(x)  =  a,    if   ()  <  .r  <  ^>i, /(.r)  =  ./.,    if 
bi  <  X  <  h.^,   f(x)  =  as    if    /',  <  .r  <  bs  ,  &C., 


i[ 


1 1  tan~^  -  +  (('i  —  ((.->)  tan~^ '-  +  (>/.,  —  a,)  tan~^ ^ 

.'/  U  .'/ 

+  (''3  —  c-i)  tan-^  — + 

1/ 


3.  Show  that  if  f(x)  =  -  1  if  x  <  —  1.  f(x)  =x  if  —  1  <  ./•  <  1 . 
f(x)  =  l    if   x>l. 

r=i[(l+.)tan-.l±^-(l-.)ta„-.l^  +  |log|l^^]. 

4.  Show  that  if  f(x)  =  -  1  if  .r  <  -  1 ,  f(x)  =0  if  -  1<  ,/■  <  1 , 
/(.T)  =  l     if     a->l, 

T-      ir        ,l±x      ^        ,l-.r~| 
ttL  //  ?/     J 

Show  that  the  equipotential  lines  are  equilateral  hyperbolas  passing  through 
the  points  ( —  1,  0)  and  (1,  0),  and  that  the  lines  of  flow  are  Cassinian  ovals 
having  (—  1,  0)  and  (1,  0)  as  foci.  The  lines  of  flow  are  equipotential  lines 
and  the  equipotential  lines  are  lines  of  flow  for  the  case  where  the  points 
( —  1,  0)  and  (1,  0)  are  kept  at  the  same  infinite  potential,  or  where  very  small 
ovals  surrounding  these  points  are  kept  at  the  same  finite  potential.  The  case 
is  approximately  that  of  a  pair  of  wires  connected  with  the  same  pole  of  a 
battery  whose  other  pole  is  grounded,  and  then  placed  with  their  ends  in  con- 
tact with  a  thin  plane  conducting  sheet. 

5.  Show  that  if  f(x)  =  0  if  x  <  0 ,  f(x)  =  —  1  if  ()<'  ,/■  <  a,  f(x)  =  0 
if  a  <x<b,    and    /(.r)  =  1    if   x  >  b, 

,.       ir-TT  <t—x  J,  —  x  ,.r~| 

/  =  —     —  —  tan~  ^  — — ■  —  tan~  ^ ■  —  tan~  ^  -     . 

TT  L  i  ii  ij  >jA 

The  conjugate  function 


-■'■f  +  ir^(J->-x^+>f] 

is  the  solution  for  the  case  where  a  sink  and  two  sources  of  equal  intensity  lie 
on  the  axis  of  A',  the  sink  at  the  origin  and  the  sources  at  the  distances  a  and 
b  to  the  right  of  the  origin.  One  of  the  lines  of  flow  is  easily  seen  to  be  the 
circle     x^  -\-  if-  =  ab . 

47.     If  the  plane  conducting  sheet  has  two  straight  edges  at  right  angles 
with  each  other  and  one  is  kept  at  potential  zero  while  the  value  of  the  poteu- 


Chap.  IV.]  EXAMPLES.  TT 

tial  function  is  given  at  each   point  of  the  second,  that  is   if      r=()     when 
x  =  ()     and      V^ /(■'')    when    y  =  0,    tlie  sohition  is  readily  obtained.     It  is 

00  cc 

F=  -  Cda  fe-^'-^/XX)  sin  ax  sin  aX.dX  .  (1) 

0  0 

V.  (9)  Art.  44. 
This  reduces  to 

V.  Ex.  3  Art.  45. 

EXAMPLES. 

1.  If    V=0    when    ?/ =  0    and    V^F(//)    when   a;  =  0    show  that 

r=  -  j  (la  i  f'-"-^  F(\)  sin  a//  sin  aA.rfA 

=  -fF(X)dX   \__^,^^^_^y  -  ^.+   (l  +  ^)j    • 

2.  If    7'=/(;r)    when   y  :=  0    and    F=^F(//)    when    ,t  =  <)    show  that 

3.  If   i^(//)  =  ^*   the  result  of  Ex.  2  reduces  to 
2/^ 


TT 


tan-  '^  +  i  f/(A)rfA  T^^^ -,  -    ,   ,     ^     ,1 


4.    If    F(!/)  =  1    for    0  <  ,y  <  1     and    FQ/)  =  0    for    //  >  1    while    f(x)  =  1 
for   0<a'<l    and    f{x)  =  0    for   ,t  >  1 

F=  iftan-  ^-^  -  tan-  ^-±^  +  2  tan-  "^ 

+  tan-  1^^  -  tan-  ^-^^  +  2  tan-  '^l  • 
X  X  //J 


78  SOLUTION    OF    PKOBLEMS    IN    PHYSICS.  [Art.   48. 

5.    If  one  edge  of  the  conducting  sheet  treated  in  Art.  47  is  insulated,  so  that 
D^.V=0    if   .r  =  0    and    V=f(x)    Avhen    y  =  0 


V=-  ( '^a  i  e-'^yf^X)  cos  ax  cos  aX.dX 


48.  If  the  conducting  sheet  is  a  long  strip  with  parallel  edges  one  of  which 
is  at  potential  zero  while  the  value  of  the  potential  function  is  given  at  all 
points  of  the  other,  that  is  if  F=0  when  y  =  (i  and  V^=F(x)  when 
y^h   the  problem  is  not  a  very  difficult  one. 

Since  we  are  no  longer  concerned  with  the  value  of  V  when  y  ^=  cc 
V=  e'^y  sin  ax  and  F=  pJ^^  cos  ax  are  available  as  particular  solutions  of  the 
equation 

i),2F+i>/r=0  (1) 

as  Avell  as    F=  e~'^y  sin  ax    and    F^  e~'^y  cos  ax  . 

,.a;/  _j_  ^-  ay 

Consequently         — sin  ax  =  cosh  ay  sin  ax         [Int.  Cal.  Art.  43  (2)] 


and — sin  ax  =  sinh  ay  sin  ax         [Int.  Cal.  Art.  43  (1)] 

and  cosh  ay  cos  ax     and     sinh  ay  cos  ax 

are  now  available  values  of  F  and  can  be  used  precisely  as     e~ °-y  cos  ax     and 
e~'^y  sin  ax     are  used  in  Art.  44. 

Following  the  same  course  as  in  Art.  44  we  get 

as  a  solution  of  (1)  which  will  reduce  to     F=  F(x)    Avhen    // =  6 
and  to  F=0     when     //  =  0,  since  sinh  0  =  — - — =0, 

and  (2)  is  therefore  our  required  solution. 

If  F  is  to  be  equal  to  zero  when   y  =  h   and  to  f(x)  when    ?/  =  0    we  have 
only  to  replace  ^  by  h  —  y   and   F(x)    by  f{x)  in  (2).     We  get 


CllAl'.  IV.]     FLOW    IN    A    LONG    STKIP    WITH    PAKALLEL    EDGES.  (  9 

If    V^f(a--)    when    y  =  0    and    V=F(x)    when    y  =  />    then 

y^l  P„  rsinha(^-^)  ^^^  _ 

ttJ       J         sinh  ah       "  ^  ^ 

0  —00 

This  can  be  considerably  simplified  by  the  aid  of  the  formula 

.       pTT 

^  .   ,  sm  — 

/smh  px              ,         TT                    (I 
.   ■■    —  cos  rx.dx  =  -—  • 
smh  ox                        1(1         j)ir   .         .    rir 
<|                                               cos h  cosh  — 

if  7^^<  '/'-.     [Bierens  de  Haan,  Tables  of  Def.  Int.  (7)  265]  and  becomes 
1     .     TT  „  .   /^.„_  d\ 


F=5^ain^(«-,„)j:/-(A)— ^^^^ 


cos  ,    ' 1-  cosh  -  (A  —  x) 

0  0  ^  ^ 


2b  b  J      ^        _  TT// 


d\ 


cos  -j-  -\-  cosh  T  (A  —  .'•) 


'cosh  —  (A  —  x)  —  cos  -f-      cosh  —  (A  —  x)  4- 
b  b  b  ^ 


cos    , 
0 


EXAMPLES. 

1.  Given  the  formula 

— r~7~ — r~  =    ,  tan- 1  (a//   ,      tanh  J )       if  6  >  ff , 

«  +  6cosha'       sjb'^  —  a-  \^b-\-a  2/  ' 

show  that  if    F=l    when    ^=^0    and    V=^Q    when   [/=^b    V=j(b  —  i/). 

2.  Show    that    if     V=^0      when     i/=^b,     V^  —  1      when      t/  =  0     and 
X  <C0  ,    and    F=  1    when    //  =  0    and    x  >  0 

TT  L    ,  TT//   J 

tan- 

The  solution  for  the  conjugate  system,  that  is,  for  a  strip  having  a  source  at 
(0, 0)  and  an  infinitely  distant  sink  is 


log     cosh^  — ^  —  cos^  -77-     ■ 

TT         L  ^b  zb  _i 


80  SOLUTION    OF    PROBLEMS    IX    PHYSICS.  [Art.   48. 

3.  Show  that  if  7'=  —  !  when  ^  =  0  and  .r<.().  r=l  when  // =  0 
and  .r  >  0 .  V^ — 1  when  y=^h  and  .r  <  0  ,  and  V^l  Avhen 
y  —  h    and   ./'>0. 

2  /         TT  7r.''\        2  /         Ti"  7r>"\ 

1'=  -  tan-i  ( tan  ^  (/'  —  y)  tanh  -77  I  H —  tan"^  I  tan  —  //  tanh  -—  I 

TT  \         Jib  Zb  /        IT  \        lb  '  2b/ 

.        TT.r 

2     ,r^^^  ^1 

=  -  tan-M  ■ 

TT  L      .       TTI/  J 

sm  —r- 
b 

The  solution  for  the  conjugate  system,  that  is,  for  a  strip  having  a  source  and  a 
sink  at  the  points  (0,  0)  and  (0,  b)  is 

TT.T  Try 

-cosh  — -  +  cos  ^  . 

r     -■     ■         *  '■ 


TT  1-  Tra'  7r//J 


4.    If    7'=0    Avhen    .r  =  0,    7'  =  /(.r)  when    //=:()    and   ,r>0,    and    V={) 
when    //  =  Ji    and    a-  >  0  , 

V=^~  Cda  p"iha(/;-//)  ^^.^^  ^^^  _  ^.^  _  ^^^  ^^^  ^  .r)]/(X)^X 
TT.'       ;;^         sinh  aJ> 

= h  -"  t/T ^ ^ 1/W''^ 

2//  ''   r       L  ,     TT^^  ^  73-//  1     TT   ..       I         X  TT'/ 

cosh  —(>^  —  .'■)  —  cos  -i-        cosh  —(AH-  .<■)  —  cos  -r- 
b  0  b  ^  b 

for  positive  values  of  x  and  for  values  of  1/  betweeen  0  and  b. 

0.    If     J'l^O     when    .t  =  0,     Vi  =  F(x)    when     y  =b    and    ./•>0,    and 
Ti  =:  0    when    ij  =  0    and   .r  >  0 

r,  =  J_  sin  -^  f  r ^ : 1 ~]F(X)dX 

.   cosh  —  (A  —  y)  +  cos  — ^       cosh  —  (A  +  x)  +  cos  -j- 

for  positive  values  of  .r  and  values  of  .y  between  0  and  b. 

6.  If     Vo  =  0     Avhen    ;r  =  0 ,     F2  =  /(•'^)     when    y  =  0     and    .*■  >  0 ,    and 
F2  =  -Z^(a:-)    when    //  =  b    and    .-:?;>  0 

7;  =7^+  7^1     for     .X-  >  0    and    ^<y<h.    (v.  Exs.  4  and  5) 

7.  If  one  edge  of  the  strip  described  in  Art.  48  is  insulated  so  that  we  have 
V=^f{x)    when   y  =  0    and   2)^7^=0    when    y^^b   show  that 


V 

TT, 


1  p^  posha(^-,/)  ^^^         _       ^^^ 

rrj       J         cosha^>       ''^  ^  ^ 


Chap.  IV.]  FLOW    OF    HEAT    IN    ONE    DIRECTION.  81 

By  the  aid  of  the  formula 


/ 


,    rir         pTT 
cosh  —  cos  -— 

COSh/JCC  TT  J,q  Jy 


COS  rx.dx  = '^ =^  if    P  <  2? ; 

ITT 

7 


cosh  qx             '           q          lyir   .         .   rir 
^  cos \-  cosh  — 


[Bierens  de  Haaii,  Def.  Int.  Tables  (6)  265] , 
reduce  this  to 

1    ,              .    ./Wco.h|(A-.) 
V=  7  sin  -y  j   ^ dX. 

'  _^     cosh  T  (■^  —  ■^j  —  cos  -y- 

8.  If  F=0  when  ^  =  0  or  Z*  and  .r<  — «,  F=l  when  y  =  0  or  b 
and    —  o<.r<a,     and    F=0    when    // =  0    or    />   and    x>  a 

.        7r(a—.r)                      .        7r(r/ +  ./•) 
smh  — ^— smh  — ^— 

F=  -     tan-i h  tan-i -' ■     • 

TT   L  Try  .       TT//  J 

sm  -J-  sm  — ^ 

9.  If  r=0  when  ?/ =  0  or  />  and  x'<  — a,  F=l  when  //  =  0  and 
—  a  <  ,r  <  « ,  F=  0  when  //  =  0  or  b  and  x  >  a,  and  F=  —  1  when 
y  =  b    and    —  ^/  <  .r  <  a 


V- 


tanh^^^^                 tanh^^^^^ 
^  [tan- ^-  +  tan- ^—1 

TT  \_  ^         Try  ^         TT?/  J 

tan  -y-  tan  -j- 


10.  A  system  conjugate  to  that  of  Ex.  9  is  F^  +  oo  when  // =  0  or  h 
and    .r  =  —  (^? ,     V=  —  cc    when    ?/  =  0    or   b   and   a'  =  a .      In  this  case 

.    .,  TT//         .        7r(^f  —  .r) 
sm-  -; — \-  smh — ^— 

2^        sin^:^^  +  sinh-^^^^^> 

49.  Let  us  take  now  a  problem  in  the  flow  of  heat.  Suppose  we  have  an 
infinite  solid  in  which  heat  floAvs  only  in  one  direction,  and  that  at  the  start  the 
temperature  of  each  point  of  the  solid  is  given.  Let  it  be  required  to  find  the 
temperature  of  any  point  of  the  solid  at  the  end  of  the  time  t. 

Here  we  have  to  solve  the  equation 

D,u  =  a'I)^^u  (1) 

[v.  Art.  1  (ii)]  subject  to  the  condition 

uz=f(x)     when     ^^  =  0.  (2) 


82  SOLUTIOX    OF    PKOIiLEMS    IN    PHYSICS.  [Art.  49. 

As  the  equation  (1)  is  linear  with  constant  coefficients  we  can  get  a  particu- 
lar solution  by  the  device  used  in  Arts.  7  and  S. 
Let    M  =  eP'  +  <"    and    substitute  in  (1).     We  get 

13  =  a-a^ 

as  the  only  relation  which  need  hold  between  /?  and  a. 

Hence  u  =  e"-^  +  ""-^"-'  =  e""-""  e'^'  (3) 

is  a  solution  of  (1)  no  matter  what  value  is  given  to  a. 
To  get  a  trigonometric  form  replace  a  by  al. 

Then  u  =  e-"'""  e"^. 

If  in  (3)  we  replace  a  by  —  at    we  get 

As  in  Arts.  7  and  8  Ave  get  from  these  values 

H  =.  e-"-"^-'  sin  ax     and     ii  =  e~"^°--*  cos  ax 

as  particular  solutions  of  (1),  a  being  Avholly  unrestricted. 

From  these  values  we  wish  to  build  up  a  value  of  u  which  shall  reduce  to 
/(a-)  when    t  =  0   and  shall  still  be  a  solution  of  (1). 

We  have  /(.r)  =  1  Cda  Cf{X)  cos  a  (A  —  x).d\  (4) 

V.  Art.  32  (3),  and  by  proceeding  as  in  Art.  44  we  get 

u  =  l  Cda  Ce-"'''"  f(X)  cos  a(X  —  x).dX  (5) 

as  our  required  value  of  u. 

This  can  be  considerably  simplified. 
Changing  the  order  of  integration 

u  =  1  ff(X)dX  fe-""'"  cos  a(X  —  x).da .  (6) 

fe-"-"-'  cos  a(X  —  x).da  =  —  ^-  .  e~  H^  (7) 

by  the  formula 

00  J 

Ce-"'^'  cos  bx.dx  =  —■  e~Si         [Int.  Cal.  Art.  94  (2)] 

0 

Hence  ic  = p^|/(A)e      T^dX.  (8) 


Chap.  TV.]  EXAMPLES.  83 

X  —  .r 
Let  now  B  = p- , 

2a\f 

then  X  =  ,r  +  2asJ}.^ 

and  u  =  ^Cf{x-{-2a^t.(3)e-^'d/3.  (9) 


EXAMPLES. 


1.    Let  the  solid  be  of  infinite  extent  and  let  the  temperature  be  equal  to  a 
constant  c  at  the  time     ^  =  0 . 


Then  it  =  ^  Ce-  ^'  dfi  =  ~  A--  ^ y//?  = . 

Int.  Cal.  Art.  92  (2). 

2.    Let     u  =  X     when     t  =  0 . 

Then  ^^  =  ^  fc^'  +  2asJll3)e-^'d/3  =  x. 


3.    Let 

?< 

=  x'^ 

when 

t  — 

:(). 

Then 

?^^  = 

■■x^ 

+ 

4.    Let 

W  : 

=  0 

if   x<- 

-l>, 

U  = 

1 

if 

hen    t  = 

0. 

Then 

h<x<h,    and   w  =  0    if   .r  > /> 


•^avt 

n=^Ce-^^,B  =  UJi ^3  +  3,,.      ,5+10.3.^  +  5^  _^_^^^ 

V^J  "^      \/7rL2r/v/^      3(2aV^^'  5.2!(2«v/^)5  J 

h  +  x 

5.    Let   «  =  0    if  .:*;<0    and   u  =  l    if   a- >  0   when   ^  =  0. 
Then 


~2      ^^L2a^t      'd.(2a^ty      5.2\(2a\Jtf      7.31(2(^0'  J' 

6.  An  iron  slab  10  c.  m.  thick  is  placed  between  and  in  contact  with  two 
very  thick  iron  slabs.  The  initial  temperature  of  the  middle  slab  is  100°,  and 
of  each  of  the  outer  slabs  0°.  Required  the  temperature  of  a  point  in  the 
middle  of  the  inner  slab  fifteen  minutes  after  the  slabs  have  been  put  together. 
Given     «^  =  0.185    in  C.G.S.  units.  Arts.,  21°. Q>. 


84  SOLUTION    OF    PROBLEMS    IN    PHYSICS.  [Art.  50. 

7.  Two  very  thick  iron  slabs  one  of  which  is  at  the  temperature  0°  and  the 
other  at  the  temperature  100°  throughout  are  placed  together  face  to  face. 
Find  the  temperature  of  each  slab  10  c.  m.  from  their  common  face  fifteen 
minutes  after  they  have  been  placed  together.  Ans.,  70°.8,  29°.2. 

8.  Find  a  particular  solution  of  D,u^  a'^D^u  on  the  assumption  that  it 
is  of  the  form  n  =  T.X  where  T  is  a  function  of  t  alone  and  X  is  a  function 
of  X  alone. 

50.  If  our  solid  has  one  plane  face  which  is  kept  at  the  constant  tem- 
perature zero,  and  we  start  \vith  any  given  distribution  of  heat,  the  problem  is 
somewhat  modified. 

Take  the  origin  of  coordinates  in  the  plane  face.  Then  we  have  as  before 
the  equation 

D,  u  =  (v^l)^:  a  ,  (1) 

but  our  conditions  are 

u  =  [)     when     x  =  0  (2) 

u=f(x)   "  ^  =  0  (3) 

and  we  are  concerned  only  with  positive  values  of  x. 
We  may  then  use  the  form  (4)  Art.  32 

f(x)  =  -  (da  I /(A)  sin  ax  sin  aX.dk ,  (4) 

and  proceeding  as  in  the  last  section  w^e  get 

u^=—  (  da  i  e'"''^-'f(X)  sin  ax  sin  aX.dX  (6) 

0  I-' 

as  our  reqiiired  solution.     This  may  be  reduced  considerably. 

II  =  -  if{X)dX  I  e-"-°--' [cos  a{X  —  x)  —  cos  a{X  +  x)'\da, 


n  =  — ^  Cf{X)  (e-  ^i;;^'  —  e-  ^-^^)dX  (6> 

bhis  may  be  reduced  to  the  form 


2a\/'7Tt;; 
by  (7)  Art.  49,  and  this  may  be  reduced  to  the  form 


2av'«  2<iv'^  ( 

EXAMPLES. 

1.    Let  the  initial  temperature  be  constant  and  equal  to  c. 


lAP.  IV.]  EXAMPLES.  85 

Then 


"=Kf'--"'"-f'^"'^'] 


2a\/t 
2(' 


\jrJ 

2^  r  X 

N/tt  \-2a\/i 


+ 


^'ir\-2a^i      3.(2as/ty      5.2l(2a\/t)'       1.3\{2<t\Jty  J 

2.  Assuming  that  the  earth  was  originally  at  the  temperature  7000°  Fahren- 
heit throughout,  and  that  the  surface  was  kept  at  the  constant  temperature  0°, 
find  (1)  the  temperature  10  miles  below  the  surface  10,000,000  years  after  the 
cooling  began;  (2)  the  temperature  1  mile  below'  the  surface  at  the  same 
epoch;  (3)  the  temperature  10  miles  below  the  surface  100,000,000  years  after 
the  cooling  began;  (4)  the  temperatiTre  1  mile  below  the  surface  at  the  same 
epoch;  (5)  the  rate  at  which  the  temperature  was  increasing  with  the  distance 
from  the  svirface  at  each  point  at  each  epoch. 

Neglect  the  convexity  of  the  earth's  surface  and  take  Sir  Wm.  Thomson's 
value  of  ci^  (400)  the  foot,  the  Fahrenheit  degree,  and  the  year  being  taken  as . 
units.      (Thomson  and  Tait's  Nat.  Phil.  Vol.    II.  Appendix.) 

Ans.,  (1)  3114°;  (2)  332°.5;  (3)1036°;  (4)  103°;  (5)  1°  for  every  20  feet,  3° 
for  every  50  feet,  1°  for  every  50  feet,  1°  for  every  50  feet. 

3.  Let  the  initial  temperature  be  constant  and  equal  to    —h,   then  by  Ex.  1 

'  V7r,v' 

4.  Let  the  temperature  of  the  plane  face  be  h  instead  of  zero,  and  let  the 
initial  temperature  be  zero. 

Then  we  have  only  to  add  b  to  the  second  member  of  the  solution  in  Ex.  3, 
as  we  may  since    n^h    is  a  solution  of  (1)  Art.  49,  and  we  get 

2aVl 
O 


■•'-'' i'-rJ''""^) 


5.    Let     u^b   when   x^O    and    u=f(x)    when    ^  =  0, 
Then 


(A  +  xf 


by  (6)  Art.  50. 


86  SOLUTION    OF    PROBLEMS    IN    PHYSICS.  [Art.  51. 

6.  Let   u^h   when   a?  =  0    and    u  =  c    when    f  ==  0 . 

Then  v  ^I,  +  (r  —  f>)  ^  fe"  ^^  dp . 

vV 

7.  If  the  earth  has  been  cooling  for  200,000,000  years  from  a  uniform  tem- 
perature, prove  that  the  rate  of  cooling  is  greatest  at  a  depth  of  about  76 
miles,  and  that  at  a  depth  of  about  130  miles  the  rate  of  cooling  has  reached 
its  maximum  value  for  all  time.     Let    a^  =  400. 

8.  Show  that  if  the  plane  face  of  the  solid  considered  in  Art.  50  instead  of 
being  kept  at  temperature  zero  is  impervious  to  heat 

u  =  -^-=  Cf(X)  (e      """-'    +  e"  ~^^^  )cl\ .  V.  (6)  Art.  50. 

51.  If  the  temperature  of  the  plane  face  of  the  solid  described  in  Art.  50 
is  a  given  function  of  the  time  and  the  initial  temperature  is  zero,  the  solution 
of  the  problem  can  be  obtained  by  a  very  ingenious  method  due  to  Riemann. 

Here  we  have  to  solve  the  equation 

I),u  =  a^D^u  (1) 

subject  to  the  conditions 

u  =  F(t)     when     x=^()    ■> 

I  (2) 

We  know  that 

X 

is  a  solution  of  (1),  v.  Ex.  1  Art.  50.     It  is  easily  shown  that 


u  =  ^  Ce-^'dfi,  (3) 

VttJ 


where  c  is  any  constant,  is  a  solution  of  (1). 
For 


i»  '» 


Vtt  2a>^t  —  c 


2  1  ^r —  r  ~2   ' 

sItt  2a^f  —  c  4«'(^  —  c)  Of^ir 

and    .  D,v  =^a^D:u. 


€hai'.  IV.]  TEMPERATURE    OF    FACE   A   FUNCTION    OF    THE    TIME.  8< 

Let     cji(_jc,  t)     be  a  function  ot  x  and  /  which   shall    be   ec|uai   tu  zero   if  t  is 
negative  and  shall  be  equal  to 

'-hf'-""' 

if  t  is  equal   to  or  greater  than   zero;    so  that   if   a'  =  ()    cp(,r,  f)  ^1    and   if 

We  shall  now  attack  the  following  problem,  to  solve  equation  (1)  subject  to 
the  conditions 

u  =  0         if     f  =  0 

n  =  F{{))    ''     X  =  0     and     0  <  /  <  r 

u  =  F(kT)''     x  =  0       "       A;t</<(;.'  +  1)t, 

where  //  is  any  whole  number  and  r  is  any  arbitrarily  chosen  interval  of  tinie. 
If  we  form  the  value 

u  =  F(kT)[<l.(x,  t  -  kr)  -  (^(.r,  /  -  (7/  +  1)t)]  (4) 

u  will  satisfy  equation  (1)  since  zero,  unity  and 


2    r 

tJ'-"-"' 

are  values  of  u  which  satisfy  (1).      u  will  be  zero  if    j'  <  kr    by  the  definition 
of  the  function    <f>(x,  t);    if   x  =  0    u  =  0    if    ;'  >  (/.■  +  1)t    and    it  =  F(kT)    if 

kT<t<{]i-+l)T. 

Therefore 

k=x 

is  the  solution  of  the  problem  stated  above. 

(5)  can  be  simplified  somewhat  from  the  consideration  that  for  a  given  value 
of  t  <f)(x,  t—  kr)  =0  if  kT>  t.  If,  then,  71t  is  the  greatest  whole  multiple 
of  T  not  exceeding  f. 

f:  =  « 

n  =  ^  F(kT)[^(x,f -  kr  )  -  c^(.r,  f  -  (k  +  1  )t)] .  (6) 

If  now  we  decrease  r  indefinitely  the  limiting  form  of  (G)  will  be  the  solu- 
tion of  the  problem  stated  at  the  beginning  of  this  article. 

(6)  may  be  written 


=  XF(kr)  ^</>(■*^^-A^r)-<^(..,^-(^-+l)r)^ 


(') 


88  SOLUTION    OF    PIIOBLEMS    IN    PHYSICS.  [Art.  51. 

and  if  r  is  indefinitely  decreased  the  limiting  form  of  (7)  is 

u  =  -  CF(\)D,<l,(.r,f  —  \)dX  .  (8) 

Since    t  —  X   is  positive  between  the  limits  of  integration 


2aV7 


2    r 
^(x,t  —  A)  =  1  -  —  j  e-^M^, 
\7rJ 


and  D,^(:.r,t  -x)  =  --^e   ^«^^'-^>  (^  -  A)". ;    , 

2«V7r 

and  (8)  may  be  written 

t  ,(2 

u  =  -^  CF(X)e~ '"'"-''  (f  -  X)~ I  dX ,  (9) 

or  if  we  let 


2a\t  —  X 


EXAMPLES. 
1.    If    /(  =  nt   when    ./■  =  0    and    u  =  0    when    t  =  0 


2.  A  thick  iron  slab  is  at  the  temperature  zero  throughout,  one  of  its  plane 
faces  is  then  kept  at  the  temperature  100°  Centigrade  for  5  minutes,  then  at 
the  temperature  zero  for  the  next  5  minutes,  then  at  the  temperature  100°  for 
the  next  5  minutes,  and  then  at  the  temperature  zero.  Required  the  tem- 
perature of  a  point  in  the  slab  5  cm.  from  the  face  at  the  expiration  of  18 
minutes.     Given;  cf^=.185.  .1/^s'.,  20°.l. 

3.  If    It  =^  F(t)    Avhen    ,r  =  0    and    ii  ^f(^v)    when    ^  =  0  ,    then 
V.  (6)  Art.  50. 


Chap.  IV.]  TEMPERATURE   A  PERIODIC    FUNCTION    OF    THE   TIME.  89 

4.  If  in  Art.  (51)  F{t)  is  a  periodic  function  of  the  time  of  period  T  it  can 
be  expressed  by  a  Fourier's  series  of  the  form 

F(t)  =  -  />„  +  ^  [«,„  sin  mat  -\-  b,j^  cos  mat']  ,     where      a  =  ^ , 

,H=    1 

or  F(t)  =  ^  bo  +  X  Pm  s"i  (">^f  +  Ki) , 

where  p„^  cos  A,„  =:  a,,^     and     p,,^  sin  A„j  =  i,„.  v.  Art.  31  Ex.  3. 

Show  that  with  this  value  of  F(t)  (10)  Art  51  becomes 

H  =  ^  bofe-^^//3  +  ^  X  P,n  [sin  (mat  +  A,J  J«-^=  cos  ^J  r/yS 

—  cos  (m,at  +  A,„)  |  f~^^  sin  "^^^    (7y8"l 
and  that  as  t  increases  u  approaches  the  value 

Given  that 

Ce—'  sin  -,  (7.^;  =  ^  e-'^^^-gin  ^  72  ;    fe-  '^  cos  -,  (/,/•  =  —  e-*-^^"  cos  b  V2. 
J  .r''  2  J  x'^  2 

0  f 

V.  Riem^ann,  Lin.  par.  dif.  gl.  §  54. 

0.  If  we  are  dealing  with  a  bar  of  small  cross-section  where  the  heat  not 
only  flows  along  the  bar  but  at  the  same  time  escapes  at  the  surface  of  the 
bar  into  air  at  the  temperature  zero  we  have  to  solve  the  differential 
equation 

DfU  =  (I'W^ii  —  b~u  .  v.  Fourier,  Heat  §  105. 

SlioAv  that  for  this  case 

a  =  e-^^-  +  "'""'>'  sin  ax     and     ;^  =  g-  "*'  +  "-^- "  cos  ax 
are  particidar  solutions,  and  that  if     k  =_/'(,/•)     when     t  =  0 

u  =  '-^  re~-^f(\)dX  =  ^'  P'-^=/(.f  +  2asft.^)dfi . 
-'^sJirtJ  sIttJ 

cf.  (8)  and  (9)  Art.  49. 


90  SOLUTION    OF    PROBLEMS    IN    PHYSICS.  [Art.  5L 

If    u^O    when    .r  =  0    ajid    u  =f(-f)    when    ^  =  0 

u  =  ~p  [_f''-^\f(^  +  2a\/f.f3)df3  -fe-P\fi-  .r  +  2a^t.f3)d(3~j  . 

cf.  (7)  Art.  50. 

If   n^=  —  e'~77    when    f^O    and    )f=^0    when    .<■  =  0 

and  if  u  =  l  when  .r  =  0  and  u  =  0  when  f  ==  0  we  have  only  to  add 
e~f  to  the  second  member  of  the  last  equation,  since  u^e"^  satisfies  the 
equation 

D,  11  =  (iW^  u  —  h-  II  . 

If     ft  =^  F(t)    when    .« =  0    and    ii  =  0    when    f  =  0    we  can  employ  the 
method  of  Art.  ol. 


2aVt 
3 


ZhStt 

and  u  =-^7=  r(^  -  X)~2e-^^('-A)-,-;;i^^  i'\A)f/A  , 

2a  VW 

cf.  (9)  Art.  51, 

2aV^« 

cf.  (10)  Art.  51. 

If  F(t)  is  periodic  and  has  the  value  taken  in  Ex.  4,  show  that  the  value 
approached  by  u  as  t  increases  is 

1  ,       '>.'■     "'^  xv/2„    .     /      ,      ■'>-s/2      ,    ^    \ 

"  =  ^  ^) ''    «  +  2^  Pm  f'-'ir ''  sm  I  mat—  -^  y  +  A,  J  , 

where  p  =  (//-  +  V'''' +  »r-a-)^     and     -/  =  (—  //^  +  V/'*  + '»"^a^)^  • 


Chap.  IV.]  ANGSTilOMS    METHOD.  91 


Given 


and 
where 


/^-      -  ^"     7  ^■^     -■',■  O  7 

t;~'-~  ,2  COS  —  f/.r  =^  -T-  «       cos  Zrt  , 


ll  (^,,2  _|_  v/,,4  ^  /,4^i     ^j,^i     ,7  =,  V2  ^_  ^^..  ^  ^^^^  ^  ^^^^1 


Angstrom's  method  of  determining  the  conductivity  of  a  metal  is  based  on 
the  result  just  given  (v.  Phil.  Mag.  Feb.  1863),  and  is  described  by  Sir  Wm. 
Thomson  (Encyc.  Brit.  Article  "Heat'')  as  by  far  the  best  that  has  yet  been 
devised. 

52.  If  II  is  a  periodic  function  of  the  time  when  ,/;  ^  0  as  in  Art.  51  Ex.  4 
and  we  are  concerned  with  the  limiting  value  approached  by  u  as  t  increases 
we  can  avoid  evaluating  a  complicated  definite  integral  if  we  take  the  following 
course.  , 

Since  as  we  have  seen  in  Art.  49     u  =  e^'  +  °-''     is  a  solution  of 

J),  II  =  ci^I);  u  (1) 

provided  only  that     /3  =  d-a-     we  have 

as  a  solution. 

Replacing  /3  by    ±  /3I   tins  becomes 

or  ir  =  e^^''^-^f^'^'^ 


v//  =  ±^n/2(1  +  o. 


and  V—  /  =  ±  7^  V2  (^  T  ''^  ■ 

Hence 


.,  =  e-t^^  sin  (l3t  -  I  yjl)  ,    „  =  ^^\  COS  {pt  -  ^  ^1)  '  (2) 

«  =  .:^lsin(/3*  +  ;^^f),      „  =  „;;V|„„,(^,  +  i^|),  (3) 


are  particular  solutions  of  (1). 


(^) 


92  SOLUTION    OF    PROBLEMS    IN    PHYSICS.  [Akt 

From  these  we  get  readily 

xJma    .     /  X     \ma   ,        \ 

"  =  /3„,^-'-„  *T  sill  {^waf  —  -\^r-  +  K,) 

as  a  solution.     (4)  reduces  to 

n  =  p„,  sin  (maf  +  A,„)     Avhen     .r  =:  0 

and  to  n  =  p„,  g-f  ^T  sin  ( A„,  —  -  a  / — )    when    ?^  =  0 . 

If  we  add  a  term  which  satisfies  (1)  and  which  is  equal  to  zero  Avhen    x-  =  0 
and  to     — /D,„^"o     2  sin  ( A,„  — -\/— ;- j     when     f  —  0     (  v.  Art.  50)  we  shall 
have  a  solution  of  (1)  which  is  zero  when     t  =  0     and  whicli  is 
p,„  sin  (mat  +  A,„)     when     x  —  O. 

The  term  in  question  approaches  zero  as  f  increases  [v.  (7)  Art.  50]  and  we 
have  at  once  the  solution  given  in  Art.  51  Ex.  4,  as  our  required  result. 

EXAMPLE. 

Show  that    t/^(^^t  +  '^-^   is  a  solution  of   J),v=z(rlJ;u  —  1/n    if   /3='>-a-  —  f/^, 
and  hence  that 


?<:^e~^siii 
where 


(f^f  ±  -^-) ,    and    »  =  e^  3l  cos  (tSt  ±  ^)  , 


p  =  [v/y3^  +  lA  +  ^>^]i     and     y  =  [v//3^  + />^  -  ^--^ji , 
are  solutions.     Hence 

"-=/>V'-;Ssin(^^--^  +  A„,) 

is  a  solution. 

If  (3  =  ma  this  last  result  reduces  to  ?«  =  p^  sin  (waf  +  A^)  when  x  =  0 
and  by  the  reasoning  of  Art.  52  it  must  be  the  value  tt  approaches  as  t  increases 
if  we  have  the  same  conditions  as  in  the  last  part  of  Art.  51  Ex.  5. 

53.  The  whole  problem  of  the  flow  of  heat  is  treated  by  Sir  William  Thom- 
son (v.  Math,  and  Phys.  Papers,  Vol.  II),  and  other  recent  writers  from  a  dif- 
ferent and  decidedly  interesting  point  of  view,  which  we  shall  briefly  sketch 
in  connection  with  the  problem  of  Linear  Flow. 

Suppose  we  are  dealing  with  a  bar  having  a  small  cross-section  and  an  adia- 
thermanous  surface,  and  take  as  our  unit  of  heat  the  amount  required  to  raise  by 
a  unit  the  temperature  of  a  unit  of  length  of  the  bar.    If  at  a  point  of  the  bar  a 


Chap.  IV.]  INSTANTANEOUS    HEAT    SOURCES.  93 

quantity  Q  of  heat  is  suddenly  generated  the  point  is  called  an  instanfrmeoiis 
heat  source  of  strength  Q. 

If  the  heat  instead  of  being  suddenly  generated  is  generated  gradually  and 
at  a  rate  that  would  give  Q  units  of  heat  per  unit  of  time  the  point  is  called  a 
jyermanent  heat  source  of  strength  Q. 

The  temperature  at  any  point  of  the  bar  at  any  time  due  to  an  instantaneous 
source  of  strength  Q  at  the  point  x  =  X  is  easily  found  by  the  aid  of  formula 
(8)  Art.  49  as  follows:  — 

If  a  quantity  of  heat  Q  is  suddenly  generated  along  the  portion  of  the  bar 
from  .r  =  A  to  .r  =  x  -(-  AA,  where  AA  is  any  arbitrary  length,  the  tem- 
perature of  that  portion  will  be  suddenly  raised  to  -^ ,  and  we  shall  have  by 
(8)  Art.  49 

u=^^J>'^dX  (1) 

as  the  temperature  of  any  point  of  the  bar  at  any  time  t  thereafter. 

If  now  we  write  u  equal  to  the  limiting  value  approached  by  the  second 
member  of  (1)  as  AA  is  made  to  approach  zero  we  get 

n  =  e     icr-t  (2) 

as  the  solution  for  the  case  where  we  have  an  instantaneous  source  at  the 
point     ./•  =  A . 

It   is   to   be  observed  that   in  (2)      v  =  0     when     ^  ^  0      and      n  =       ' 


2nsJ'Trf 
when    X  =  A    and    fX). 

If  we  have  several  sources  we  have  only  to  ad  1  the  temperatures  due  to  the 
separate  sources. 

Formula  (8)  Art.  49  may  now  be  regarded  as  the  solution  for  the  case  where 
we  start  with  an  instantaneous  heat  source  of  strength  f(X.)dk  in  every 
element  of  length  of  the  bar. 

A  source  of  strength  —  Q  is  called  a  sink  of  strength  Q;  and  (6)  Art.  50 
may  be  regarded  as  the  solution  for  the  case  where  we  have  at  the  start  an 
instantaneous  source  of  strength  /(A)cZA  in  every  element  of  the  bar  whose  dis- 
tance to  the  right  of  the  origin  is  A ,  and  an  instantaneous  sink  of  strength 
/(X)dX  in  every  element  of  the  bar  whose  distance  to  the  left  of  the  origin  is  A. 

If  we  have  an  instantaneous  source  at  the  origin  (2)  reduces  to 

(''  _Zl  /ON 

2iiSTt 


94  SOLUTION    OF    PROBLEMS    IN    PHYSICS.  [Art.  54. 

For  a  permanent  source  of  constant  strength  Q  at  the  origin  (3)  gives 

u  =  -^  (  e-  i^^iTT^)  (f  -  r)- 2  dr  (4) 

2a\7rJ 

and  for  a  permanent  source  of  variable  strength  f(f) 

u  =  —-=  Ce-^^^,^  (t  —  T)-if(T)dT  .  (5> 

2a\7rJ 

In  (4)  and  (5)  u  obviously  reduces  to  zero  when  t  =  0  and  ./■  >  0 ,  but  its 
value  when  a-  =  0  is  not  easily  determined.  We  can  avoid  the  difficulty  by 
introducing  the  conception  of  a  doublet. 

54.  If  a  source  and  a  sink  of  equal  strength  Q  are  made  to  approach  each 
other  while  Q  multiplied  by  their  distance  apart  is  kept  equal  to  a  constant  P 
the  limiting  state  of  things  is  said  to  be  due  to  a  doublet  of  strength  F  whose 
axis  is  tangent  to  the  line  of  approach  and  points  from  sink  to  source.  A 
doublet  of  strength  —  P  differs  from  a  doublet  of  strength  P  only  in  that  its 
axis  has  the  opposite  direction. 

Let  us  find  the  temperature  due  to  an  instantaneous  doublet  of  strength  P 
placed  at  the  origin.  For  a  source  of  strength  Q  at  .v  —  rj  and  an  equal  sink 
at    X  =  —  f)    we  have 


(J  (7)-J-)2  (r?  +  x)2 

(6        4a2<  6        4a2<    ) 


or  if   2r]Q  =  P, 


2a>f7ri 


P  _  (r)2  +  x2)  rfX  n^ 

j=e  4n2«       (e2a2t  —  e     2a".l) 

■ia-qSTTt 

P  (t)2  +  t2)     .  -nx 

-=  e      i^ST"  smh  -^  . 

2arj>^'Trt  ^a^ 


If  7}  is  made  to  approach  zero 


"[^'"''II]=l4' 


limit 

-V 

and  u  = p=  e-^,  (1) 

is  the  solution  for  the  temperature  at  any  time  and  place  due  to  an  instantane- 
ous doublet  of  strength  P  placed  at  the  origin.  For  a  doublet  at  any  other 
point   a-  =  X   we  have 

P(X  —  X)  (.r-A)2  ^. 

4«V7r^^ 


if   .r  >  0 ,    and 


Chap.  IV.]  PERMANENT    DOUBLET.  95 

For  a  penuanent  doublet  of  constant  strength  P  placed  at  the  origin  we 
have 

0 

and  for  a  permanent  doublet  of  variable  strength  f(t) 

^'  =  7^7=  re-i:fc)  {i  -  ryKf(r)dr  ,  (4) 

a   x<  0,    if  we  let    /3  = y^^  • 

From  (5)  and  (6)  Ave  see  readily  that  u  =  0  Avhen  ^  =  0  and  that 
M  ='77^     when     *■  =  0    if  we  approach  the  origin  from  the  right  and  that 

u  =  —^TT^    when    x  =  0    if  we  approach  the  origin  from  the  left. 

If  the  point  a-  =  0  is  kept  at  the  constant  temperature  b  and  we  are  con- 
cerned only  with  positive  values  of  x  we  can  get  from  (5)  the  solution  given  in 
Art.  50  Ex.  4  by  supposing  a  permanent  doublet  of  strength  2a^b  placed  at 
the  origin. 

To  solve  the  problem  treated  in  Art.  51  we  have  only  to  suppose  a  permanent 
doublet  of  strength  2a^F(t)  placed  at  x  =  0  and  from  (5)  we  get  at  once 
(10)  Art.  51. 

EXAMPLE. 

Show  that  if  D^u  =  aW^u  —  li-u  and  an  instantaneous  source  of  strength 
Q  is  placed  at   x  ^  \ 

Q        h->,  (•^  -  ^)-  .      -^  1^    ^ 

u=^ -,:=- e   "■'      4„2(  V.  Art.  ol,  F,x.  o. 

2a\7rt 

Show  that  if  an  instantaneous  doublet  of  strength  P  is  placed  at  the  point 
_     Px       _,,_^ 


96  SOLUTION    OF   FliOBLEMS   IN   PHYSICS.  [Art.  55. 

If  a  permanent  doublet  of  strength  f(t)  is  placed  at   x^=0 


-Jj-^f'-^'i'-^)"'' 


2a>/'t 

f(f) 

whence  t(  =  0    when  ^=^0  and  .r  >  0  or  ,r  <  0  and  ii^=--±'-~     when 

'Ji/- 

x  =  0. 

Hence  if  we  place  at  ;?"  =  0  a  permanent  doublet  of  strength  2a'^F(f)  we 
get  the  solution  given  in  Art.  51  Ex.  5  for  the  case  where  u  =  F(t)  when 
.r  =  ()  and  t/  =  0  when  ^  =;  0  provided  we  are  concerned  only  with  positive 
values  of  x . 

If   F(t)  =  c   this  reduces  to 

2r,   /^  b2x^ 

u  =  -=  I  e-^-~i^2  (113 . 
\7rJ 


2aV'7 

55.  As  another  example  of  the  use  of  Fourier's  Integral  Ave  shall  consider 
the  transmission  of  a  disturbance  along  a  stretched  elastic  string. 

Suppose  we  have  a  stretched  elastic  string  so  long  that  we  need  not  consider 
what  happens  at  its  ends,  that  is  so  long  that  we  may  treat  its  length  as 
infinite.  Let  the  string  be  initially  distorted  into  some  given  form  and  then 
released  ;  to  investigate  its  subsequent  motion. 

Let  us  take  the  position  of  equilibrium  of  the  string  as  the  axis  of  X  and 
any  given  point  as  origin. 

We  have,  then,  to  solve  the  differential  ecjuation 

I)f!jz=a^D^l/  (1) 

[v.  (vin)  Art.  1]  subject  to  the  conditions 

1/  =f{x)  when  t  =  0  (2) 

I),>/  =  0  "      t  =  0.  (3) 

As  in  Art.  8  we  find 

1/  =  cos  a(x  ±  at)    and    y  =  sin  «('^'  i  at) 
as  particular  solutions  of  (1). 

From  these  we  must  build  up  a  value  that  will  reduce  to 

f(x)  =  -  Cda  Cf(X)  cos  a(X  —  x).dX  (4) 


Chap.  IV.]  INFINITE    STEP^TCHED    ELASTIC    STRING.  97 

when    f^O    and  will  at  the  same  time  satisfy  (3). 

//  =  cos  aX  cos  a(.r  -\-  at)  -{-  sin  a\  sin  a(.r  -\-  at) 
or  //  =  cos  a(X  —  .>•  —  at) 

is  fi  solution  of  (1). 


Hence 


//  =  -  Cda  Cf(k)  cos  a(\  —  .r  —  at).d\  (5) 


is  also  a  solution  of  (1). 

(5)  reduces  to    //  =/(.t)    when    t  =  0    but  it  gives 

i)^7/  =  —  Tr/a  CflX.)  sin  a  (A  —  ,r).d\ 

when    t  =  0    and  consequently  does  not  satisfy  equation  (3). 

If  in  forming  (5)  we  use     cos  a(x  —  at)     and     sin  a(.r  —  at)     instead  of 
cos  a(.r  +  at)     and     sin  a(.r  -f-  at)    we  get 

?/  ==  -  Cda  C/IX)  cos  a  (A  —  ,r  +  at).dX  (6) 

which  is  a  solution  of  (1),  and  reduces  to   //=,/'(>)    when    t  =  0  ,    l)ut  it  gives 

Dtl/  =  —  —  Cda  Cfl\)  sin  a(A  —  .r).d\ 

w^ren     t^O     and  does  not  satisfy  (3). 

If,  however,  we  take  one-half  the  sum  of  the  values  of  //  in  (5)  and  (6)  we 
get 


/O.^/A 


+  -  Cda  Cf(\)  cos  a  (A  —  ,r  +  ^^O-r/A   | . 


(") 


a  solution  of  (1)  which  satisfies  both  (2)  and  (3),  and  is,  therefore,  our  required 
solution. 

This  result  can  be  very  much  simplified. 

If  we  substitute     ,-;  =  x  -{-at 

-  Cda  ("/(A)  cos  a(\—x  —  at).d\ 
=  -  Cda  Cf(\)  cos  a(X  -  z).dX  =./U)  =/(.r  +  .^0  ; 

TT./  »/ 


98  SOLUTION    OF    PROBLEMS    IN    PHYSICS.  [Art.  5(5. 

and  in  like  manner  we  can  show  that 

-  Cda  Cf(X)  cos  a(A  —  x  +  at).dX  =f(x  —  at)  . 

0  —00 

Hence  our  solution  becomes 

>/=lLf(:'-+"0+f{-'--'^t):\-  (8) 

This  result  is  of  great  importance  in  the  theory  of  elastic  strings  and  it 
shows  that  the  initial  disturbance  splits  into  two  equal  waves  which  run  along 
the  string,  one  to  the  right  and  the  other  to  the  left,  with  a  uniform  velocity  a, 
and  that  there  is  nothing  like  a  periodic  motion  or  vibration  of  any  sort  unless 
the  ends  of  the  string  produce  some  effect. 

56.  If  the  string  is  not  initially  distorted  but  starts  from  its  position  of 
eqiylibrium  with  a  given  initial  velocity  impressed  upon  each  point  we  have  to 
solve  the  equation 

U^>/  =  aW^l/  (1) 

subject  to  the  conditions 

//  =  ()     when     t  =  0  (2) 

D,y  =  Fi^x)-         f-0.  (3) 

We  get  by  the  process  used  in  Art.  55 

=  :^Jf(X),JX  j^pin_a^:X-.  +  ^^0  _  ^n.  a(X-.r  -  nt)y^  , 


but 


*^  a  '^       .  a 

it,     anc 


if     X  —  at  <.X<.  X  +  at,     and  is  equal  to  zero  for  all  other  values  of  A;  since 
"^•^7.,._      -  if  "'>0 


=      0   if  m  =  0. 
V.  Int.  Cal.  Art.  92  (3). 

x  +  at 

Hence  y  =  ^CF{X)dX  (4) 

X  —  at 

is  our  required  solution. 


Chap.  IV.]  LONG    RECTANGULAR    PLATE.  99 

EXAMPLES. 

1.  If  the  string  is  initially  distorted  and  starts  with  initial  velocity  so  that 
y=_/'(,r)     and    I),j/=F(.r)    when    ^  =  0 

v  =  \  UV  +  "^)  +/(•'■  -  "0]  + 1;  J^X^)^^  • 

2.  If  the  initial  disturbance  is  caused  by  a  blow,  as  from  the  hammer  in  a 
piano,  which  impresses  upon  all  the  points  in  a  portion  of  the  string  of  length 
c  an  equal  transverse  velocity  b  show  that  the  front  of  the  wave  which  will  be 
seen  to  run  to  the  left  along  the  string  will  be  a  straight  line  having  a  slope 

equal  to     —      and  a  length  equal  to     —  v'4''/-  +  Ifi  .     Of  course  a  wave  having 

a  front  of  the  same  length  with  a  slope  equal  to     —  —     will  be  seen  to  run  to 

the  right  along  the  string,  and  the  effect  of  the  two  waves  will  be  to  lift  the 

string  bodily  and  permanently  to  a  distance    —    above  its  original  position. 

57.  We  shall  now  take  up  a  few  examples  of  the  use  of  Fourier's  Series. 
In  the  problem  of  Art.  7  let  the  temperature  of  the  base  of  the  plate  be  a 

given  function  of  x,  the  other  conditions  remaining  unchanged. 

Since  f(x)  =  ^  (a„^  sin  m x) 

»i=  1 

a    =:  —  I  f(a)  sm  ma.da 

IT  J 

?f  =  —  ^     ^^~"'''  sin  mx  I  f{a)  sin  ma.da     .  (1) 

If  the  breadth  of  the  plate  is  a  instead  of  ir 

"=J,X\j     "   ''"  "7"  J  •  W  sin -7^  f^^  J  •  (2) 

58.  If  the  temperature  of  the  base  is  unity  and  the  breadth  of  the  plate  is 
TT  the  solution  is,  as  we  have  seen  in  Art.  7, 

?f  =  -      e"-'  sin  x-\-  -  e~  •''■"  sin  3.r  -|-  -  er'^"  sin  5x  +  •  •  •      .  (1) 

7r  L  o  5  A 

This  series  can  be  summed  without  difficulty.     We  have  the  development 
log(l  +  .)  =  ^-f  +  |-^+--- 
if  the  modulus  of  z  is  less  than  1.      Int.  Cal.  Art.  221  (4). 


where 


we  have 


100  SOLUTION    OF    PROBLEMS    IN    PHYSICS.  [Art.   58. 

Hence  log  (1  —  -)  =  — 


-,  -,2  ^3  .A 

if  mod.  .-  <  1 . 

1 


and 


if  mod.  .i  <  1 . 
But 


[log  (1  +^)  -log  (1  -^j]  =1  +:^  +^  +  ...  (2> 


log  (1  +  z)  =  log  [1  +  >'(eos  <^  +  ;  sin  <^)] 

=  I  log  [(1  +  ;•  cos  <^)-^  +  (>■  sin  </,)'^]  +  /  tan-^  .  ™  ^ 
-  1  -f-  /'  cos  <f> 

=  I  log  (1  +  2r  cos  ck  +  >)  +  /  tan- 1  .    ' '"'  ^  ^  , 

J  ^  1  -j-  /■  cos  cfi 

and 

log  (1  —  --■)  =  -  log  (1  —  2r  cos  <f>  +  >'-)  —  /  tan-^     ^'^!"  ' 

[Int.  Cal.  Art.  33  (2)], 
and  (2)  becomes 


1  ri  -      1  +  2y  cos  cf>  +  r'       .  .  2r  sin  ^H 

2  L2  ^^^  l-2>-cosc^  +  .-  +  '  ^'-^'^     T^J 


^cos  <^  +  /  sin  <^)       >-^(cos  3<^  +  >'  sin  3<^) 

i  *"  3 


(3) 


From  (3)  we  get  two  equations 


1  ,      1  +  2r  cos  <^  +  /•- >'  cos  cf>       r^  cos  3<^       /-^  cos  5c^  . . 

4  ^^  1  -  2r  cos  <^  +  r'  ~  ~^        ^  3  ^         5  ^  ^^ 

1  ,        ,2/'  sin  d)       r  sin  <i    ,    r^  sin  3<i    ,    r^  sin  o<i   ,  ._s 

5tan-^-^^=-^  +  — 5— +-^— +  ••■  (6) 

both  valid  for  all  values  of  <^  provided  /■  <  1 . 
e~"  is  less  than  1  if  3/  is  positive. 
Hence  from  (5) 

e-''sin.r       «"■'•"  sin  3.r   ,   f^"-"*"  sin  5.r   ,  1  ,2p~''sina? 

^:— +— 7 —  +  ^5 —  +  •••==  2''^"   T^T--^ 

1  ^        T  2  sin  ./'        1  ^        ,   sin  x 
=  -  tan"  ^  — — - —  =  -  tan^ 


sinh  jj 
and  (1)  may  be  written 

sinh  // 


Ohap.  IV.]  STATIONAIIY    TEMPEJtATUUE.  ^^Jl 

If  we  replace  r  by  er"  and  (f>  by  .r  in 

log  [1  +  /'(cos  <^  +  /  sin  <^)] 
it  becomes  -  log  [1  +  e"''  cos  x  +  '  ''"""  sin  .r] 

or  log  [1  +  cos  z  +  i  sin  z] 

V.  Int.  Cal.  Art.  35  (3)  and  (4) 
a  function  of  «  as  a  whole;  and 

log  [1  —  7'(cos  (f)  +  I  sin  <^)] 

becomes  log  (1  —  cos  .-;  —  i  sin  ,-) ; 

hence  by  Int.  Cal.  Arts.  209  and  210, 

1  ,      1  +  2e-'-'  cos  cc  +  e--^  ,      1  ,  %r "  sin  x 

T  h)g  :; -; j T      and      -  tan~^  — -r- 

4     ^  1  —  2e-^  cos  £c  +  e--^  2  1  —  e--" 

1 ,      cosh  1/  +  cos  X  .      1  ^        ,  sin  a; 

or  -  log z-^ and      -  tan  ^  -^-j —  - 

4        cosh  t/  —  cos  X  2  smh  y 

are  conjugate  functions,  and 

1  ,      cosh  ?/  +  cos  X  .^. 

«i  =  -log ^-^^ (0 

TT        cosh  ^  —  cos  X 

is  the  solution  for  the  problem  where  the  isothermal  lines  are  the  lines  of  'flow 
of  the  present  problem  and  the  lines  of  flow  are  the  isothermal  lines  of  the 
present  problem.  * 

For  our  problem,  then,  the  isothermal  lines  are  given  by  the  eqviation 


o 


tan 


sni  X 


IT  sinli 


or  -^-j —  ==  tan  -—  («> 

smli  //  2 

and  the  lines  of  flow  by 


1 ,      cosh  >/  +  cos  X 

—  log  7-^ :=  b  , 

IT  cosh  //  —  COS  X 

cosh  If  +  cos  X 

r-^ =  e'^6  . 

cosh  u  —  cos  X 
EXAMPLES. 


(9) 


1.    If     D;n  -\-  D^ii  =  0  ,    and     »  =  1     when     [/  =  0  ,     and  u  =  0    when 
=  0    and  when    x  =  a  , 

4  r        wj,      .       TTX     ,     1  ■ATT,,      .       OTT.J'     ,     1          ,577,,      .       57rX  ~\ 

a^—  \  e    a  sm r  7^  ^  »  sm \-  i^  &     «  sm [-  ■  • 

TT  L                 (to                    a         o                    a  J 


2 

-  tan~^ 


vr  .   ,   Try 

smh  -^ 


l-K'-l  SOLUTION    OF    PHOBLEMS    IX    PHYSICS.  [Akt.  59. 

1'.    Ji'     f -- <fi(^j)     wlieii     //  =  0,     "  =/(//^    when     x^^O,     aud     it=- Fi^y) 
when    ./•  =  (I 

?/  =^  -  >  ^'"  ~r  sm I  <^(A)  sm dX 

+  i  sin  !^-  fr 1 ^ 1/-(A)rfX 

2(1  (I  J    \_         ,     TT  TT.*-  ,     IT  .  77./;  ^ 

(I       cosh  -  (A  —  If)  —  COS  —      cosh  -  (A.  +  //)  —  cos  —  -" 


+ r^  -"  -  rr — ' '- >(«"^  ■ 

TT    "-cosh  -  (A  —  v)  +  COS  —      cosh  -  (A  +  y)  +  cos  — 
V.  Art.  48.  Exs.  4,  o,  and  6. 

59.  If  three  sides  of  a  plane  rectangular  sheet  of  conducting  material  be 
kept  at  potential  zero  and  the  value  of  the  potential  function  at  every  point  of 
the  fourth  side  be  given;  to  find  the  value  of  this  potential  function  at  any 
point  of  the  sheet. 

To  formulate:  — 

D;r+i)/r=o.  (i) 

V=i)    when    ./■  =  ().  (2) 

7'=0         "         :r  =  a.  (3) 

7'=0        -        y  =  h.  (4) 

y=f{^)   "        2/  =  <'-  (5) 

Working  as  in  Art.  48  we  get 

sinh  — -  (/)  —  //) 

'I  .     mirx 

, sm 

.    ,    imrli  a 

smh 

a 

as  a  value  of  F  which  satisfies  equations  (1),  (2),  (3),  and  (4)  if  m  is  an  integer. 
Therefore 

F=  -  y    -—  sm I  /(A)  sm dk  (6} 

„,  =  1  smh " 

a 

is  our  required  solution. 


Chap.  IV.]  RECTANGULAR  PLATE.  108 

EXAMPLES. 
1.    If   f{x)  =  1    Eq.  (6)  Art.  59  reduces  to 

sinh  -  (A  —  ij)                      sinh  —  (J>  —  //) 
4r          «.                    .     TT.r    ,1             ((                     .     Sttx 
I  —-\   -, sm ho T—, sm 

TT  L  .     1     TTO  a  6  .     .     OTTO  (l 

siiin  —  smn 

«  a 

smh  —  {(>  —  >/) 
+  i ^^ sm 1 


2.  If    r=0    when    x  =  0,    V=0    when    cc  =  a,    V=0    when    ?/  =  0, 
and    F  =  F(x)    when    //  =  b ,    then 

m=co    smh a 

F==  -  V     r  sm (  FCk)  sm d\     • 

a  -^  I     •   1   ^"  ''■'^  «    J      ^  a  J 

7n=i    smh 0 

a 

3.  If    i^(cc)  =  I    the  answer  of  Ex.  2  reduces  to 

.    ,   irii  .   ,  Stt?/  .   ,  Stt?/ 

smh  -^  smh  — ^        ^  smh  — -         „  _ 

_^  4  r  ^/         .        TTX      ,1  O         .      OTT.X     ,     1  a         .      OTTX     ,  \ 

'  =  - 7  sm ho   5-7  sm h  i Tl  sm • 


"sinh  —  sinh ■  sinh 

a  a                                         a 

4.    If    r=0    when   x=^0,    F=0    when    x=za,    V^f(x)    when    y  =  0, 

and    V^F(x)    when  y^h,    then 

m  =  Qo  gml^  /^  _  ^A      a 

F=  -   >,     sm { I  f(\)  sm dX 

m  =  1  smh t> 


smn "  V       V  -. 


sinh 


5.    If  f(x)  =  F{x)    the  answer  of  Ex.  4  reduces  to 


.cosh 


7i^  /  A  _     \ 

^  ii    V2       ^/     .     niTTX  r^^.     .    mirX  ,  H 

=  -  >, sm I  f(X)  sm dX 

a''~i\_  .   vnrb  a    J'  ^  '  a  J 

cosh  — - —  0 

2a 


104  SOLUTION    OF    PROBLEMS    IN    PHYSICS.  [Art.  59. 

6.  If  /(.r)  =:  F[.r)  =  1    the  answer  of  Ex.  5  reduces  to 

V=z-  \ sill h  .. :r—, ^"^ 

TT  L  ,   -TTh  a         3  ,   S-rrb  « 

cosh  -—  cosh  — — 

cosh^-  (^  —  I/)         .  _, 

+  ^ ^^ sm . 

O  ,    OTTO  a  J 

cosh  — — 
2a 

7.  If     r=/(a')    wlien     //  =  0,     V=F(x)     when    // = /^     r=<f)(i/)     when 
a;  =  0,    and    V=x(>/)    when    x=za,    then 

smh  — -  (b  —  If) 


r=-X      sm ( ^ )/(A)sin- 

m=\  smh " 

a 

.   .  miry 
smh "  ^  _, 


sinh 

a 


smh  — —  (ft  —  x) 


sinh     , 
h 


.    ,   m.iTX 
smh 


+  -7^>)^^"T'"^)] 


sinh     , 
b 

8.    If   /(.t)  =  <^(^)  =  0    and   i'^(a')  =  x(//)  =  1    the  answer  of  Ex.  7  may  be 
reduced  to 

sinli  7  (7;  —  .^' )  .  cosh  ^  (-  —  .*• ) 

V=-\  TTT- sm  ^  +  o :;^ sm  -t^ 

TT  L  26  .    ,   TTff  6        2  ,   lira  b 


sinh—  cosh   2^ 

3  .   ,   37ra  6         4  .   4,7ra 

smh^_  cosh  — 


47r//  "1 

b  J 


Chap.  IV.]  FLOW    OF    HEAT    IN    A    SLAB.  105 

9.  Find  the  temperature  of  the  middle  point  of  a  thin  sqviare  plate  whose 
faces  are  impervious  to  heat;  1st,  when  three  edges  are  kept  at  the  tem- 
perature 0°  and  the  fourth  edge  at  the  temperature  100°;  2d,  when  two 
opposite  edges  are  kept  at  the  temperature  0°  and  the  other  two  at  the  tem- 
perature 100°;  3d,  when  two  adjacent  edges  are  kept  at  the  temperature  0° 
and  the  other  edges  at  the  temperature  100°.     See  examples  3,  6,  and  8. 

A71S.,  (1)  25°;   (2)  50°;   (3)  50°. 

60.     Let  us  pass  on  to  the  consideration  of  the  flow  of  heat  in  one  dimension. 

Suppose  that  we  have  an  intinite  solid  with  two  parallel  plane  faces  whose 
distance  apart  is  c. 

Take  the  origin  in  one  face  and  the  axis  of  X  perpendicular  to  the  faces. 
Let  the  initial  temperature  be  any  given  function  of  x  and  let  the  tAvo  faces  be 
kept  at  the  constant  temperature  zero;  to  find  the  temperature  at  any  point  of 
the  slab  at  any  time. 

We  have  to  solve  the  equation 

I),n  =  r>^D^u  (1) 

subject  to  the  conditions 

;,  =  0    when    ,r  =  0  (2) 

;^  =  0       "       ./•  =  «  (3) 

u  =f(x)  '•'        ;-  =  0 .  (4) 

In  Art.  49  we  have  found 

u  =  6~  """■'  sin  ax 

and  i(  =  e-"-'^-'  cos  ax 

as  particular  solutions  of  (1). 

{(=6-"^'^''  sin  ax     satisfies  (2)  Avhatever  value  is  given  to  a.     It  satisfies  (3) 

if    a  = provided  m  is  an  integer.     Let  us  try  to  build  a  value  of  ii  out  of 

terms  of  the  form    Ae-  "  "U     sin  — ^     whieli  shall  satisfy  (4). 
We  have 

/(.,.)  =  fX  [sin  ^^//'(X)  sin  =-\/x].  (S) 

»i  =  1  U 

?<  =  -  2^     e -r-  sm  —-^  J  /"(A)  sm  ——  <1X  \,  (6) 

n,  =  1  0 

reduces  to  (5)  when    t  ^Q    and  is  our  required  solution. 


106  SOLUTION    OF    PROBLEMS    IN    PHYSICS.  [Art.   61. 

EXAMPLES. 

1.  If    f{X)  —I>.    a  constant,  (6)  Art.  60  reduces  to 

4:b  r      (r-n"-!    .     ir.r       1       >M"-n''-t    .     Sttx   .    1       257^^1    .     Bttx    ,         ~\ 

u  =  —     e"  c2    sm [-  7;  c~    c2    sm \-  ~  i~  ^^  sm h  ' "  '     • 

TT  L  c         6  CO  (■  J 

2.  An  iron  slab  10  cm.  thick  is  placed  between  and  in  contact  with  two 
other  iron  slabs  each  10  cm.  thick.  The  temperature  of  the  middle  slab  is  at 
first  100°  throughout,  and  of  the  outside  slabs  0°  throughout.  The  outer  faces 
of  the  outside  slabs  are  kept  at  the  temperature  0°.  Eequired  the  temperature 
of  a  point  in  the  middle  of  the  middle  slab  fifteen  minutes  after  the  slabs  have 
been  placed  in  contact.     Given  ^2  =  0.185  in  C.G.S.  units.  Ayis.,  10°.3. 

3.  Two  iron  slabs  each  20  cm.  thick  one  of  which  is  at  the  temperature  0° 
and  the  other  at  the  temperature  100°  throughout,  are  placed  together  face  to 
face,  and  their  outer  faces  are  kept  at  the  temperature  0°.  Find  the  tem- 
perature of  a  point  in  their  common  face  and  of  points  10  cm.  from  the  com- 
mon face  fifteen  minutes  after  the  slabs  have  been  put  together. 

Arts.,  22°.8;   15°.  1;   17°.2. 

4.  One  face  of  an  iron  slab  40  cm.  thick  is  kept  at  the  temperature  0°  and 
the  other  face  at  the  temperature  100°  until  the  permanent  state  of  tem- 
peratures is  set  up.  Each  face  is  then  kept  at  the  temperature  0°.  Required 
the  temperature  of  a  point  in  the  middle  of  the  slab,  and  of  points  10  cm.  from 
the  faces  fifteen  minutes  after  the  cooling  has  begun. 

Ans.,  22°.8;  15°.6;  16°.T. 

61.  If  the  faces  of  the  slab  treated  in  Art.  60  instead  of  being  kept  at  the 
temperature  zero  are  rendered  impervious  to  heat,  the  solution  of  the  problem 
is  easy. 

In  this  case  we  have  to  solve  the  equation 

subject  to  the  conditions 

l)^u  =  {)    when    .r  =  0 

y,  =/(.r)  "         ^  =  0. 
We  have  only  to  use  the  particular  solution 


as  we  used  u  =  e   "-°--'  sin  ax 

in  Art.  60.     We  get 

u  =  I  [^pWrfX  +X(.-'^  COS  '^pm  cos  '^  <IX)  ]  .  (1) 


Chap.  IV.]    FLoW    OF   HEAT  IN  A  SLAB  WITH  ADJAUATIC  FACES.  107 

EXAMPLES. 

1.  Solve  example  2  Art.  60  supposing  that  the  outer  surfaces  are  blanketed 
after  the  slabs  are  placed  together  so  that  heat  can  neither  enter  nor  escape. 
Find  in  addition  the  temperature  of  the  outer  surfaces  fifteen  minutes  after 
the  slabs  are  placed  in  contact.  Ans.,  33°.3;  33°.3. 

2.  Solve  example  3  Art.  60  on  the  hypothesis  just  stated,  getting  in  addition 
the  temperatures  of  points  on  the  outer  surfaces. 

Ans.,  50°;  33°.9;  66°.l;  27°.2;  72°.S. 

3.  Solve  example  4  Art.  60  supposing  that  heat  neither  enters  nor  escapes 
at  the  outer  surfaces  after  the  permanent  state  of  temperatures  has  been  set 
up.     Find  also  the  temperatures  of  points  in  the  outer  surfaces. 

Ans.,  50°;  39°. 7;  60°.3;  35°.5;  64°.;5. 

4.  Show  that  if  ;^^=  0  when  .r  =  0,  D^.u^O  when  x  =  e,  and  u=f(x) 
when    ^  ^  0 , 


(2m  +  l)7rA  ^^^> 


"  =  ,  S(' "^^ '"'  ^ 2,         j  /(^) 

Suggestion:  Assume  u=^()  when  x  =  2<i  and  /(2c  —  .v)=zf(x),  and  see 
(6)  Art.  60. 

62.  If  the  temperature  of  the  right-hand  face  of  the  slab  considered  in  Art. 
60  is  a  constant  y  instead  of  zero  we  have  only  to  add  to  the  second  member 
of  (6)  Art.  60  a  term  u^  which  shall  satisfy  the  conditions 

I),  u,  =  (i-D^  n,  (1) 

„^  =  0    Avhen    r  =  0  (2) 

n,  =  {)        •'         f  =  0  ,               (3) 

"i  =  y     ."       x  =  r.  (4) 

7ti=^     obviously  satisfies  (1),  (2),  and  (4);  to  make  it  satisfy  (3)  as  well 

we  must  add  a  term  ii.^  which  shall  be  equal  to  zero  when  x  =  0    and    when 

x'=c    and  to    —  —    when    ''  =  0,    while  always   satisfying  (1).      It  is  given 

immediately  by  (6)  Art.  60  and  is 

2y  ^  /     m'-av-t    .    mirx  r      .     itiirX      \ 


/ 


X  sm dX  = cos  niir  =  (—  1)"*+  ^  > 

c  niir  ^  mir 


108  SOLUTION    OF    PEOBLEMS    IN    PHYSICS.  [Art.  63. 

and  ?/.,  =  -^yl  ^ ^  e ^^  siii )  •  (o) 

TT  ■^  \      m  c    / 

Hence  ?;j  =  y     _  -j-  -  >   /  :^^- — ^  g-    ^2     sm  — —  )     •  (' ) 

If  the  left-hand  face  of  the  slab  considered  in  Art.  60  is  to  be  kept  at  a 
constant  temperature  jS  and  the  right-hand  face  at  the  temperature  zero  we 
can  get  the  term  n^  which  must  be  added  to  the  second  member  of  (6)  Art.  60 
by  replacing  y  hy  /3  and  x  l)y  c  —  a:  in  (7).     We  then  have 


r— ■--'"f(i.-^si„=^-)i 

L       C  TT  -^^  \iH  C      /  _| 


(8) 


EXAMPLES. 

1.    Show  that -if     n  ^  (3   when    .7- ^  0  .    v  =^  y    when    x  =  c,    and    i'=f(x) 
when    t=^0 


+  f  X(.-=^'  .sin  '^fim  -ffl  sin  '^  ,/a). 

m  =  \  n 

2.    ShoM' that  if     if  =  fi   when    x  =  i) ,    u  =  {)    when    f  =  0,    and    D^^h^O 
when    X  ^=  c 

=  /?     1 {(-  4.-2   sm  —-  -j--e-  ^^  sm h  t:  «~   4c2     sm  — [-•••)      • 


63.  If  the  temperature  of  the  right-hand  face  of  the  slab  just  considered  is 
a  function  of  the  time  instead  of  a  constant  and  the  temperature  of  the  left- 
hand  face  is  zero  the  problem  can  be  solved  by  a  method  nearly  identical  with 
tliat  of  Art.  51. 


Chap.  IV.]  TEMPEHATUllE    OF    0:NE    FACE    VARIABLE.  109 

Let  (ti(x,t)  be  a  function  of  x  and  t  which  shall  be  zero  if  f  is  less  than  zero 
and  shall  be  equal  to 


fi—l)'"      vfl,r-irh    .     mirx^ 
—  e — ';r~  sin 


[v.  (7)  Art.  62]  if  t  is  equal  to  or  greater  than  zero.      So  that 

<^(.r,0  =  0  if  t  <  0 

<^(.T,^)=0  "  ^  =  0     unless     x  =  r. 

^(x.f)  —  1  "  f  =  0  and     ,r  =  c 

<}>(x,f)  =  1  "  ,r  =  r 

<^(.r,0  =  0  ''  ;r  =  0  . 
Precisely  as  in  Art.  51  we  get 

A- = (I  *- 

as  the  required  solution  of  our  problem,  n  being  as  in  Art.   51   the  largest 

integer  in  -  where  t  is  any  given  value  of  the  time. 

On  our  hypothesis  the  last  term  of  (1),  that  is,   —  F(nT)(^[jx,t  —  (n  +  1)t]  =  0; 
the  next  to  the  last  term    F(^nr)c})(x,f  —  vt)    has  for  its  limiting  value 

while  as  in  Art.  51  the  limiting  value  of  the  rest  of  the  sum  is 
-CF(\)D;,<t>(x,f-X),/K. 

A*(.,,  t-x)  =  ^-^"X[(- 1)°'"--  ^'"-"  -■•  T]  ■ 


Henci 


"=-»[r+;X(^'-T)] 

-^  Zi\i—  1)'"'"^  sm  -^J  F{k)  e ^  <'-^>  dk\  , 


110  SOLUTIOX    OF    PROBLEMS    IN    PHYSICS.  [Art.  63. 

^  T,  ^  ,  -x-^r( — 1)'"  •   "^"'•^  / 7-1^  N 


If  we  substitute     f3  = -^ —  (f  —  A)     we  get 


« = f  ^(0 + S^"  -'  "^  ('■">  -f'-'  i'  -  i:^')'"^)]  •  <^> 


EXAMPLES. 


1.  If  the  temperature  of  the  left-haud  face  is  a  function  of  t  and  the  tem- 
perature of  the  right-hand  face  is  zero  and  the  initial  temperature  is  zero 

"  =  (i-f>(0-g[isi.T(-(0-/-'-('-^»]- 

2.  If  the  temperature  of  the  left-hand  face  is  a  function  of  f,  the  initial 
temperature  is  zero,  and  the  right-hand  face  is  impervious  to  heat 

^    ,        4.^-\r       1  .     ('2in  4-  l)7r.v  /  ^  . 

3.  If  in  Arts.  60-63  we  are  dealing  with  a  bar  of  small  cross-section  and  of 
length  c  and  heat  is  radiating  from  the  surfaoe  of  the  bar  into  air  at  the  tem- 
perature zero  so  that  I>,u  ^a^D;v  —  IJ^u,  show  that:  (a)  the  second  mem- 
bers of  (6)  Art.  60  and  (1)  Art.  61  must  be  multiplied  by  e~^'^  \  (h)  equation 
(7)  Art.  62  becomes 

.   ,  bx 
ismh—  m=i  '  ) 


Chap.  IV.]    VIBRATION   OF  A  STRING    FASTENED    AT    THE    E:SDS.  Ill 

(/')  equation  (2)  Art.  63  becomes 


siuli 


~  Fit)  +  3„VX  {,4+,i,,v'  ="  —  L^» 


sinh 
a 


F^/«-'*^"f^'"->nA),/A]} 


64.  The  problem  of  the  motion  of  a  finite  stretched  elastic  string  of  length 
I  fastened  at  the  ends  and  distorted  at  first  into  some  given  curve  j/^=f[.r)  , 
and  then  allowed  to  swing,  has  been  treated  and  partially  solved  in  Art.  8. 

The  complete  solution  is  easily  seen  to  be 

l/  =  j2^sin  —j-  cos  —j—^  ./(A)  sm  —j-  dX .  (1) 

in=l  0 

The  second  member  of  (1)  is  a  periodic   function  of  i  having  the  period 

21 

—.  The  motion,  then,  unlike  that  in  the  case  of  an  infinite  string  (Art.  55)  is 
*  2/ 

a  true  vibration,  a  perio(Jic  motion.  The  period  ~  is  the  time  it  takes  a  dis- 
turbance to  travel  twice  the  length  of  the  string  (v.  Art.  55). 

A  careful  examination  of  (1)  will  show  that  the  actual  motion  is  a  good  deal 
like  that  in  the  case  considered  in  Art.  55.  The  original  disturbance  breaks 
up  into  two  waves  one  of  which  runs  to  the  right  until  it  reaches  the  ^nd  of 
the  string  and  is  then  reflected,  and  runs  back  to  the  left  or  the  under  side  of 
the  string,  while  the  other  wave  runs  to  the  left  and  is  reflected  at  the  left- 
hand  end  of  the  string  and  runs  back  to  the  right  under  the  string  and  is 
again  reflected,  runs  back  to  the  left  over  the  string  and  so  on  indefinitely. 

If  the  curve  into  which  the  string  is  distorted  at  the  start  is  of  the  form 

y=^b  sin  — y^     the  solution  is 

,    .     w^TT.T         imrat  ,_, 

//  =  h  sm   --—  cos  — - —  •  (2) 

Xo  matter  what  value  t  may  have  the  curve  is  always  of  the  form 

.    .     viirx 
y^A  sm  —J- ; 

that  is,  for  different  values  of  t  we  have  a  set  of  sine  curves  differing  only  in 
the  amplitude  and  not  at  all  in  the  period  of  the  curve.  In  this  case  either 
the  whole  string  if  m  =  l,  or  each  mth  of  the  string  if  m  is  not  equal  to 
one,  rises  and  falls,  and  there  is  no  apparent  onward  motion.  When  this  is 
the  case  we  are  said  to  have  a  stead//  vibration. 


112  SOLUTION    OF    PROBLEMS    IN    PHYSICS.  [Aim.  Hi. 

If  m  =  1  we  get  steady  motion  of  the  string  as  a  Avhole  and  if  the  vibration 
is  rapid  enough  to  give  a  musical  note  the  note  is  said  to  be  the  pure  funda- 
mental note  of  the  string.  If  vi  =  2  the  vibration  is  twice  as  rapid  as  when 
w  =  I ,  the  middle  point  of  the  string  does  not  move  and  is  called  a  node,  the 
two  halves  of  the  string  are  in  opposite  phases  of  vibration  at  any  instant,  and 
the  note  given  is  an  octave  higher  than  the  fundamental  note  and  is  called  its 
pure  first  harmonic. 

If    ,11  =  3    the  vibration  is  three  times  as  rapid  as  in  the  first  ease,  there  are 

1  21  . 

tAvo  nodes   a.' =  -    and    .r  =  —  ,    and  the  note   is   the   pure   second  harmonic  of 
o  o 

the  fundamental  note. 

For  any  value  of  m  the  vibration  is  m  times  as  rapid  as  when   m  =1,  there 

are  ut  —  1  nodes  at  the  points    .r  =  —  ,  .r  ^  — ,  •  ■  •  x^ /,  and  we  get  the 

jii  ni  m 

m  —  1st  harmonic  of  the  fundamental  note. 

It  is  clear  from  (1)  that  no  matter  what  the  original  form  of  the  string  the 
resulting  vibration  can  be  regarded  as  a  combination  of  steady  vibrations  each 
of  which  alone  would  give  the  fundamental  note  of  the  string  or  one  of  its 
harmonics,  and  that  the  complex  note  resulting  is  really  a  concord  of  the  fun- 
damental note  and  some  of  its  harmonics. 

A  finely  trained  ear  can  often  recognize  in  a  complex  note  the  fundamental 
note  of  the  string  and  some  of  its  harmonics  and  is  capable  of  analyzing  a 
complex  note  into  its  component  pure  notes  precisely  as  Fourier's  Theorem 
enables  us  to  analyze  the  complex  function  representing  the  initial  form  of  the 
string  into  the  simpler  sine-functions  which  must  be  combined  to  form  it. 

EXAMPLES. 

1.    Show  that  if  a  point  whose  distance  from  the  end  of  a  harp  string  is 

-th  the  length  of  the  string  is  drawn  aside  by  the  player's  finger  to  a  distance 

11 

b  from  its  position  of  equilibrium  and  then  released,  the  form  of  the  vibrating 
string  at  any  instant  is  given  by  the  equation 


2hn'       -^  /  1      .     mTT    .     mirx         m,'jrat\ 

TT~^ Zr (  ~ s^^^  —  s^^^  ~r~ c^s  — 1~) 

n  —  IjTT- -^^  \/»'  n  I  I     / 


Show  from  this  that  all  the  harmonics  of  the  fundamental  note  of  the 
string  which  correspond  to  forms  of  vibration  having  nodes  at  the  point 
drawn  aside  by  the  finger  will  be  wanting  in  the  complex  note  actually 
sounded. 


Chap.  IV.]    VIBRATION    OF  A  STRING    IN  A   RESISTING    MEDIUM.  113 

2.  If  a  stretched  string  starts  from  its  position  of  equilibrium,  each  of  its 
points  having  a  given  initial  velocity,  so  that  we  have 

1/  =  0      when      ;'  =  0 

D,;/  =  F(x)    "  f  =  0 

y  =  0  "  .r  =  I  , 

the  solution  of  the  problem  of  its  vibration  is  easy  and  gives 

»( =  x  / 

2^/1     .     viiTX    .     m7r((t  f  ^^.     .     viirX  ,  \ 

y = ~X{-,  «"^  —  «^^^  -j-j  ^(^)  ^"^  -T  ""V  ■ 

3.  Write  down  the  solution  for  the  case  where  the  string  is  initially  dis- 
torted and  each  point  has  a  given  initial  velocity. 

65.  If  we  do  not  neglect  the  resistance  of  the  air  in  the  problem  of  the 
vibration  of  a  stretched  string  the  differential  equation  is  rather  more  compli- 
cated and  the  solution  is  not  so  easily  obtained.  The  equation  is  given  as  (ix) 
Art.  1. 

Let  us  solve  the  problem  for  the  case  where  there  is  no  initial  velocity. 

Here  we  have                     Df>/  +  2kD,u  =  <i'^D^y .  (1) 

y=z{)     when     x^O  (2) 

y^O        "         x  =  l  (3) 

y^f(x)    "           ^  =  0  '                   (4) 

B.y^O         "           t  =  0.  (5) 

We  get  particular  solutions  of  (1)  in  the  usual  way.  Assume  y  =  ^<^'+P' 
and  substitute  in  (1).     We  have 

^•■2  +  2k(S  =  a'a" 
as  the  only  necessary  relation  between  /?  and  a.     This  gives 


l3  =  ~-k±^a^a^-+k^. 
Hence  y  =  e  «'■ " ''  * '  '^«'"'  +  ^'  (6) 

is  a  solution  of  (1)  no  matter  what  the  value  of  a. 

To  throw  it  into  Trigonometric  form  replace  a  by  at,  and  since  in  actual 
problems  k,  which  is  proportional  to  the  resistance,  is  very  small,  take  —  1 
out  as  a  factor  of  the  radical.     We  have 


114  SOLUTION    OF    PROBLEMS    IN    PHYSICS.  [Art.  (55. 

Since  a  may  be  positive  or  negative  we  can  get 

(J  =  e^^*  sin  {ax  ±  t  \a-a/-  —  A:-) 


and  y  =  e~^^  cos  {ax  ±  t  \(rx-  —  //-) 

as  solutions  of  (1),  or  by  combining  these 

//  =  e~^'  sin  ax  cos  t  ya^^a"  —  h^  (J) 


y  =  e~^'  sin  ax  sin  t  \a^a^  —  k^  (8) 


y  =  e'~'-''  cos  ax  cos  f  ya'^a^  —  k^  (9) 

y  =  e~^''  cos  ax  sin  f  slcra^  —  k^  (10) 


(7)  and  (8)   satisfy  (1)   and  (2)  for  all  values   of  a.     They  satisfy   (3)  if 
^  — r- .    Let  us  sei 
(4)  and  (5)  as  well. 


a  =  —— .    Let  us  see  if  out  of  them  we  cannot  build  up  a  value  that  will  satisfy 


reduces  to  (11)  when  t  =  0   and  therefore  satisfies  (4). 


m  =1  !• 

When    t  =  0    the  first  line  of  •  the  second  member  of  (13)  vanishes  but  the 


second  line  reduces  to 

2k  x-v  /  .     mTTx  /•  ,^  ,     .     ihttX 


We  must,  then,  introduce  into  (12)  an  additional  term  which  shall  equal  zero 
when  ^  =  0  and  whose  derivative  with  respect  to  t  shall  cancel  the  term  above 
when    t  =^0. 


Chap.  IV.]    VIBRATION    OF   A   STKING   IN    A   RESISTING    MEDIUM.  115 

This  is  easily  seen  to  be 
4  '      X     ,    ,   ,  ,  sm  -—  sm  t  aP^  -  li\  I  /(A)  sm  -—  dX  . 

Hence  our  complete  solution  is 


sm  t 


\  —fr. ^  )  sm  -y- J  ,/( A)  sm  —^  dX     .     (14) 


Here  the  fact  that  e~^*,  which  decreases  rapidly  as  t  increases,  is  a  factor  of 
the  whole  second  member  shows  that  the  amplitude  of  the  vibration  rapidly 
decreases. 

Comparing  this  solution  with  that  given  in  Art.  64  for  the  case  where  there 
is  no  resistance  we  see  that  the  period  of  any  given  term 


—  cos.^,^, 


A  sm  — ;—  cos  f  \l Jv  . 


is  greater  than  tliat  of  the  corresponding  term  A^  sin  — ~  cos  ———   in  Art.  64. 

In  other  words  the  effect  of  the  resistance  of  the  air  is  to  flatten  some- 
what each  component  part  of  the  note  given  by  the  string.  More  than  this 
since  the  periods  of  the  different  terms  of  (14)  are  no  longer  exact  submultiples 
of  the  period  of  the  first  term,  the  component  notes  are  no  longer  in  perfect 
harmony  with  the  fundamental  note  of  the  string,  and  the  ideal  perfect  har- 
mony between  the  fundamental  note  and  its  harmonics  is  not  quite  realized  in 
any  actual  case. 

When  k  is  very  small,  as  in  the  case  of  a  fine  string,  the  departure  from 
perfect  harmony  is  very  slight;  but  in  the  case  of  a  coarse  string  or  worse  still 
of  an  elastic  ribbon,  where  the  resistance  of  the  air  is  considerable,  the 
unmusical  character  of  the  sound  is  very  noticeable. 

EXAMPLES. 

1.  Solve  Ex.  1  Art.  64  allowing  for  the  resistance  of  the  air. 

2.  Solve  Ex.  2  Art.  64  allowing  for  the  resistance  of  the  air; 

m  =  cc  , — ^— ^ ! 


116  SOLUTION  OF  PKOBLEMS  IX  PHYSICS.  [Art.  66. 

3.  Find  a  particular  solution  of  (1)  Art.  65  on  the  assumption  that  it  is  of 
the  form  //=T.A',  where  T  is  a  function  of  t  alone  and  A' a  function  of  x 
alone. 

66.  We  pass  on  now  to  a  couple  of  problems  that  require  the  modification 
and  extension  of  Fourier's  Theorem,  the  cooling  of  a  sphere  in  air,  and  the 
ribration  of  a  stretched  rectannidar  membrane,  but  as  an  introduction  to  the 
former  we  shall  first  consider  the  following  very  simple  problem;  to  find  the 
temperature  of  any  point  of  a  sphere  whose  initial  temperature  is  any  given 
function  of  r  the  distance  of  the  point  from  the  centre,  and  whose  surface  is 
kept  at  the  constant  temperature  b. 

Here  we  are  to  solve 

D,{rii)  =  a^D'^{ru),  (1) 

see  [v]  Art.  1,  subject  to  the  conditions 

u=f(r)    when    f  =  0  (2) 

u  =  b  '•,'  =  €  .  (3) 

if  c  is  the  radius. 

Let    r  =  rn,   then  our  equations  become 

I),v  =  aWlv  (4) 

V  =  rf(r)  when  t  =  0  (5) 

v  =  bc  "       r  =  c  (6) 

r  =  0  ''       r  =  0.  (7) 

<  )ur  problem  is  now  precisely  that  of  Art.  62  and  we  have  as  our  solution 

2^/     mV-n^'     .     m'irr  r-             .     mirX      \ 
ru  =  -  >l6      c2      sin I  \f{\)  sm dX  I 

m—l  (I 

EXAMPLES. 

1.  If  /(;■)  =  b  (8)  Art.  66  reduces  to  n^b  and  there  is  no  change  of 
temperature. 

2.  If  the  initial  temperature  is  constant  and  equal  to  /3 

1    I    -''  .o       7N  r     --%    •     -^^      1      ^-'^%    •    27r>- 


+  3e-^'sm^; J. 


Chap.  IV.]  COOLING    OF    A    SPHERE    IN    AIR,  117 

3.  An  iron  sphere  40  cm.  in  diameter  is  heated  to  the  temperature  100^ 
centigrade  throughout;  its  surface  is  then  kept  at  the  constant  temperature  0°. 
Find  the  temperature  of  a  point  10  cm.  from  the  centre,  and  find  the  tem- 
perature of  the  centre,  15  minutes  after  cooling  has  begun.  Given  (r  =  0.1Sr} 
in  C.G.S.  units.  Ans.,  2°.l;  ;r.3. 

67.  If  instead  of  having  the  temperature  of  the  surface  of  the  sphere 
constant,  the  sphere  is  placed  in  air  which  is  kept  at  the  constant  tem- 
perature zero,  the  problem  is  much  more  complicated.  For  in  this  case  the 
surface  temperature  can  no  longer  be  simply  expressed  but  is  given  by  a  ucav 
differential  equation 

D^  ic  -\-  h  II.  =  0    when     r  =  c  ,  (1) 

where  h  is  an  experimental  constant  depending  upon  what  is  called  the  sur- 
face conductivity  of  the  sphere. 
Our  equations,  then,  are 

D,{ru)  =  a^DXru)  (2) 

^t'=f{r)     when     ^  =  0  (3) 

D/ii-\-  hu^Q     when     r=^c.  (4) 

As  in  Art.  66  let    /;  :=  rii ;    then  we  have 

I),v  =  a-D;v  (5) 

V  =  rj\)')    when    ;■  =  0  (6) 

v  =  0  "        r  =  0  (7) 

D,r  +  (h  —  ^)  "  =  0    when     r  =  c.  (8) 

y?  =  g-"^"^' cos  ar  and  /?  =  e~"'""' sin  ar  have  already  been  found  as  par- 
ticular solutions  of  (5)  (see  Art.  60). 

V  ^  e~  "'"■'' iiiw  ar  (9) 

satisfies  (7)  for  all  values  of  a. 

Substitute  this  value  of  v  in  (8)  and  we  have 

ac  cos  ac  +  (lie  —  1)  sin  ac  =  0.  (10) 

If  Uj.  is  a  value  of  a  which  is  a  root  of  the  transcendental  equation  (10) 

v^e-  "'v'  sin  a^r  (11) 

will  satisfy  (5),  (7),  and  (8). 

It  remains  to  see  whether  out  of  terms  of  the  form  given  in  (11)  we  can 
build  up  a  value  of  v  which  will  satisfy  (6). 


118  SOLUTION    OF    PROBLEMS    IX    PHYSICS.  [Art.  67. 

"When  /  =  0  the  second  member  of  (11)  reduces  to  sina^.r.  If  then  we 
can  express  \/(^")  as  a  sum  of  terms  of  the  form  1/f.ii'ui  a^.r  where  a^.  is  a  root 
of  (10) 

*'  =  5)  ^'i-- ''~  "'''^'  '^^^^  "-k ''  (1- .^ 

will  satisfy  all  of  the  equations  (5),  (6),  (7),  and  (8),  and  will  be  the  required 
solution. 

Here,  then,  we  have  a  new  problem  analogous  to  that  of  developing  in  a 
Fourier's  Series,  but  rather  more  complicated,  namely,  to  develop  any  function 
of  .T  in  a  series  of  the  form    ^o„j  sin  a„j.-y    where  a„j  is  a  root  of  the  equation 

(11);  or  if  Ave  call    ar  =  </>    and    he  —  1  =7^    Avhere    a,„  =  — ^,    <^,„  being  a  root 

of  the  equation 

<^  cos  4>  -\-  p  sin  <^  =  0  (13) 

or  more  simply  of 

<^+yy  tan  </>  =  ();  (14) 

remembering  that  the  series  and  the  function  must  be  equal  for  all  values  of  x 
between  zero  and  c. 

If  <^^  is  a  root  of  (14)  —  <^,„  is  also  a  root. 

Since    sin  —^  x  =  —  sin  | '"  .r  )    the  terms  of  the  required  development 

which  correspond  to  negative  roots  may  be  combined  with  those  corresponding 
to  positive  roots,  and  therefore  we  need  consider  only  positive  roots. 

^  =  0  is  a  root  of  (14)  but  as  sin  0  ^=  0  there  will  be  no  corresponding 
term  in  the  development. 


If  we  construct  the  curve 


and  the  curve 


y=^--x  (15) 


y  =  tan  X  (16) 

the  abscissas  of  their  points  of  intersection  are  values  of  x  which  satisfy 
-  +  tana-  =  0,     that  is,  are  roots  of  equation  (14).     It  is  easy  to  see  that 

there  will  always  be  an  infinite  number  of  real  positive  roots,  one  for  each  of 
the  branches  of  the  periodic  curve  i/  =  tan  x  which  lie  to  the  right  of  the 
origin.  The  numerical  values  of  these  roots  can  be  obtained  by  an  easy  com- 
putation.    The  construction  suggested  above  shows  that  as   in  increases  <^^ 

will  rapidly  approach  the  value  (2»i  —  1)  —    if  ])  is  positive  or  if  2^  is  negative 

and  numerically  less  than  unity,  and  (2?»  -\- 1)  —  if  j^  is  negative  and  numer- 

ically  greater  than  unity. 


Chap.  IV.]  COOLING    OF    A    SPHERE    LN    AIR.  119 

There  exist,  then,  an  infinite  number  of  positive  real  roots  of  </>  +i>  tan  <^  =  0 
and  consequently  of 

ac  cos  ar  +  (//'•  —  1)  sin  ac  =  0  . 

68.  The  development  called  for  in  the  last  article  can  be  obtained  very 
easily  from  a  simpler  one  which  we  shall  now  consider,  namely,  to  develop  f(x) 
into  a  series  of  the  form 

f(x)  =  (1-^  sin  <^i,r  +  <ii  sin  </)o.r  +  <h,  sin  ^kz-v-  +  *  '  •  (1) 

where    <^i,  </>.,,  <^a  •  •  •    are  roots  of  the  equation 

^  cos  <^-{-  j>  sin  </)  =  0 ,  (2) 

the  development  to  hold  good   for  all  values  of  x  between    .r  =  0    and   ,t  =  1  . 
Let  us  proceed  as  in  Arts.  24  and  27.     Call    — r^  =  \x    and  form  it  equa- 
tions  by  substituting  for  x  in  turn  in  the  equation 

f(x)  =  (ii^  sin  <^i.'r  +  a.j  sin  t^.,./'  +  <i.  sin  c^g.''  +  ■  •  ■  +  <(„  sin  <^„.r  (3) 

the  values  Ax,  2Aa;,  SAx,  •  ■  •  nAx  ;  this  being  equivalent  to  making  the  values 
of  the  sum  and  the  function  coincide  for  the  n  values  of  x  substituted. 

To  determine  any  coefficient  «„,  multiply  the  first  equation  by  A.i'.  sin  ((^,„A.x), 
the  second  by  Ax.  sin  (2<^^Acc),  the  third  by  A.r.  sin  (3<^„,A.«),  and  so  on,  the 
nth  equation  by  A.t.  sin  (?i<^„jA.T) ;  add  the  equations  and  compute  the  limit- 
ing values  of  the  terms  of  the  resulting  equation  as  7i  is  indefinitely  increased. 
This  as  in  Art.  24  is  seen  to  be  equivalant  to  multiplying  (2)  by  sin  <^„^x.dx 
and  integrating  between  the  limits    x^=0    and   x  =  1. 

The  first  member  of  the  resulting  equation  is 


The  coefficient  of  a^.  is 


and  of  a„,  is 


I  f(x)  sin  <^„,,r.f/.x  ; 
I  sin  (f>/.x  sin  0,„.r.r/.e  , 

0 

J"sin^>,„, 


x.dx 


120  SOLUTION   OF    PKOBLEMS   IN    PHYSICS.  [Art.  68. 

I  sin  <^^.ic  sin  (f)^x.dx  =  7,  i  [cos  (<^^.  —  <^,„)a'  —  cos  (<^^.  +  (li„^)x^dx 
=  1  Fsin  (<^,  —  c^,„)  _  sin(<^,.  +  «^J~1 

^         <i>k  COS  4>k  Si^^  ^m  —  ^,n  Sm  «^^.  COS  (ji^ 

But  4>k  COS  <^^.  -{-p  sin  <^^.  =  0 

and  <^„,  cos  <^„,  +p  sin  </>,„  =  0         by  (2). 

Hence  the  numerator  of  the  second  member  of  (4)  is  zero,  and  the  coefficient 
of  a^.  vanishes  if  k  is  not  equal  to  ))i. 


fsin^  ci>,„x.dx  =  ^  [<f>,„  -  sin  <^,„  cos  <^  J  =  ^  fl  -  '^^^1 '  (5) 

2 

sin  2<^,, 


2  r 

Therefore  «„,  = ^^ — ^r—-  I  f(x)  sin  c{>„^x.dx .  (6) 


1 

The  coefficient  of  the  integral  in  (6)  can  be  transformed  as  follows  so  as  not 
to  involve  trigonometric  functions. 

^ni  cos  <^,„  +  J)  sin  <^,„  =  0 ,        by  (2) 
<^,„  cos'^  <^„,  +  tJ  sin  2<^,„  =  0  , 


sin  2c^,„ cos^</),„ 

2<^,„  F~" 

<^m'cOs2<^,„=^>2gin2^^^ 
(<^,„2+^2)cos2<^^=_p2, 


(7) 


(8) 


7^  <I^J+P' 

Hence  by  (7)  and  (8) 

sin2«^^^</>„r+/>0>  +  l) 

and  «.  =  ^l^p^^;^  sin  ^„^a.da .  (9) 

Therefore  our  required  development  is 


Chap.  IV.]  COOLING  OF  A  SPHERE  IN  AIR.  121 

From  (10)  it  easily  follows  that  for  values  of  x  between  0  and  c 

/(■'')  ^^  ''i  si^  ^1*'  +  ^'2  sin  a^x  +  ('3  sin  a^x  +  •  •  •  (11) 


'^here 


"«  =  f  ,,"+j;f+l)PW  -n  -..MIX,  (13) 

and  a,„  is  a  root  of  the  equation 

ac  cos  ar  +  p  sin  ac  =  0  .  (13) 

It  is  to  be  observed  that  if  p  is  infinite  (13)  reduces  to     sina6'=:0,     a„j 

becomes  —^  and  (11)  and  (12)  give  our   regulation  Fourier  sine  series  (v.  Art. 

31),  and  therefore  the  ordinary  Fourier  development  in  sine  series  is  merely  a 
special  case  of  the  problem  just  solved. 

Moreover  since  the  Fourier  method  of  .determining  the  coefficients  of  such  a 
series  requires  that 


I  sin  a„^x  sin  a,iX.dx  ^=  0  , 


that  IS  that  sin  (a, -Of  _  sin  (a    +  a>  _  ^ 

a«  —  a«  «,«  +  «» 

T      .         ^,    ,  a„,r  cos  a..,c       a„r,  cos  a„c 

or  recuicmg,  that  -^. —  =  -^ ~  , 

sm  a„/!  sm  a„e 

or  that  a„j  and  a„  should  be  roots  of  the  equation 

ac  cos  ac 


—. =p 

sm  ac 

Avhere  ^j  is  some  constant,  it  follows  that  we  have  obtained  in  (11;  the  most 
general  sine  development  that  can  be  obtained  by  Fourier's  method. 

EXAMPLES. 
1.    Show  that  the  solution  of  the  problem  of  Art.  67  is 


•u  =  ^^'',«e~"^V'  sin  a,„r  , 


T  ,         2      a,ff2+(Ac— 1)2    r   .    .    . 

'''''''  '-  =  c-  aP+^(A.^J^^">"""-'-^ 

and  a„  is  a  root  of 

ac  cos  ac  +  (Ac  —  1)  sin  ar  =  0 . 


122  SOLUTION    OF    PROBLEMS    IN    PHYSICS.  [Art.   (J8. 

2.  If  the  initial  temperature  of  the  sphere  is  constant  and  equal  to  /3 

ru  =  X  ^'m ''  "'"'"'  si^^  <^m  '■ 

wliere  b„  =  2/3A-     .,"'    .\^ Vx ^ 

^  2/3Ar     [a,^r--'  +  (he  -  l)-^]7 

3.  If  the  temperature  of  the  air  is  a  constant  y  instead  of  zero  the  surface 
equation  of  condition  is 

Z>,,  u  -{-  h  (u  —  y)  ^  0    when     r  =  e . 

The  substitution  of   u^  =  v  —  y,    however,  brings  the  problem  under  Ex.  2 
and  we  get 


where 


i"~y)=X 


b„,e~"''^m  sin  a^r 


'- = r  „TfT*ji„-z:ij/^[/W  -  V]  -"  "..M^- 


4.  An  iron  sphere  40  cm.  in  diameter  is  heated  to  the  temperature  100° 
centigrade  throughout;  it  is  then  allowed  to  cool  in  air  which  is  kept  at  the 
constant  temperature  0°.  Find  the  temperature  at  the  centre;  at  a  point  10 
cm.  from  the  centre;  and  at  the  surface;  15  minutes  after  cooling  has  begun. 

Given     «2  =  0.185     and    ^' =  ^     in  C.G.S.  units,     (v.  Ex.  3,  Art.  66.) 

Ans.,  96°.46;  96°.16;  95°.26. 

5.  Show  that  if  in  the  slab  considered  in  Art.  60  one  face  is  exposed  to  air 
at  the  temperature  zero,  so  that  we  have  DtU  =  a^D^u,  «  =  0  when  x  =  0, 
n=f(x)   when    t  =  0,    and    D^.)/  -\-  /n/  =^0    when   x^c,    then 

w  =  ^a^e"  "'">«'  sin  a„^x 

m  =  l 

where  «m  =  2  —„ — r^i-Ti — r^r  I  tW  sin  a,„  X.dX, 

a-c-i-  Jl(/tr-\-l)J' 
a,„  being  a  root  of     ac  cos  ae  +  /^c  sin  ae  =  0 . 


Chap.  IV.]  TEMPERATURE    OF    AIR    VARIABLE.  123 

<■>.  If  in  the  problem  of  Art.  57  heat  escapes  from  one  side  of  the  plate  into 
air  at  the  temperature  zero  so  that  we  have  I);  ii  -\-  />,;  u  =  0  ,  ?f  =  0  when 
x=^0,    ti^f{.c)    when    // =  0  ,    and    D_j.ii  -\-  hu  ^0    when   ,r  =  a,    then 

u  ==  ^''m^^"'"''  sin  a„,.« 


where 


'>n  =  2  -^-^^^_  j:/'(A)  sm  a„Ml^  , 


a„j  being  a  root  of     aa  cos  aa  +  ha  sin  a«  ^  0  . 

7.  If  in  the  problem  of  Art.  59  there  is  leakage  at  one  side  of  the  sheet  so 
that  we  have  D;V -\- Dp^=(),  F  =  0  Avhen  x  =  {),  F  =  0  when  t/^h, 
V^=f(x)    when    i/  =  0,    and    Dj,V -\-  hV—  0    when   x  =  a,    then 


sinh  a,,^^ 
where  «,„  has  the  valne  given  in  Ex.  6. 

69.  If  we  have  an  infinite  solid  with  one  plane  face  which  is  exposed  to  air 
at  the  temperatures  U  =  F(t)  and  heat  can  flow  only  at  right  angles  to  this 
face,  we  can  solve  the  problem  readily  for  the  case  where  the  initial  tem- 
peratures are  zero.     We  have 

I),v  =  aWlu 
subject  to  the  conditions 

u  =  0    when    ;'  =  0 


and  Dj.v-\- ]t(^U  —  ii)  =  Q   when   a-  =  0. 

Let  i 

Then  v  will  satisfy  the  equation 


Let  v  =  u  —  J  D^u.  (1) 


and  we  shall  also  have     v  =  U  when  x  =  0 , 


Since    U=FCt)  '- ^/^"^^^(^  -  J^^^/^  '  (2) 

by  Art.  51  (10). 


Hence 
V.  Int.  Cal.  §  4,  page  314, 


D_^u  —  hii  =  —  hv       by  (1). 
ue-''-'  =  —  h  fe-''-^  vdx  +  C ; 


124  SOLUTION    OB^    PROBLEMS    IX    PHYSICS.  [Akt.  70. 

Determining  C  by  the  fact  that    ue~''-''  =  0    when    .r  =  cc    we  have 

u  =  he'''''  Ce-'^'vdx  .  (3) 

Substituting  the  vahie  of  r  from  (2)  we  have 


as  our  required  solution. 

For  an  extension  of  this  method  to  the  flow  of  heat  in  two  and  three  dimen- 
sions and  for  the  interpretation  of  the  results  by  the  aid  of  the  theory  of 
Images,  see  E,  W.  Hobson,  Proc.  Lond.  Math.  Soc,  Vol.  XIX. 

EXAMPLES. 

1.    If  the  temperature  of  the  air  is  a  periodic  function  of  the  time,  say 
p,„  sin  (uiaf  +  A.„,)    and  we  care  only  for  the  limiting  value  of  u  as  t  incre 
show  that  this  value  is 


^ip,»'''V^ 


r/,    ,1     jma\    .     I  X     \ma   ,        \ 

1     Ima         /  X     \ma   ^        \~\ 


Art.  52  and  Art.  51  Ex.  4. 


C        .     ,      ,         e'"'(aQU].hx  —  bcoshx) 

that  I  '""  sm  hx.ax  = ^ .,  .   ,., 

J  «-  +  b- 

s  hx  + 
a'  +  Ir 


,                                  r              ,      ,         €"''(((  cos  hx  4- b  sin  bx) 
and  I  e"""  cos  bx.dx  =  — ^^ 


V.  Int.  Cal.  Table  of  Int.  (235)  and  (23t3). 

2.  If  D^.V-^D;jV=Q>,  V=0  when  ?/  =  0  and  D^^V-\- hlF{y) —  V^  =  0 
when   a?  =  0    show  that 

V=  —  f  e-"-^  dx  CF(>.)dX  f-rn-^ y,  -    o   ,    ;^    ,      J  ; 

TT  J  ^  L.r-+(A  — ^)^      a-+(A  +  //)-J 

V.  Art.  47  Ex.  1. 

70.  The  solution  for  an  instantaneous  heat  source  of  strength  Q  at  the 
point  X  ^  A  if  heat  escapes  at  the  origin  into  air  at  the  temperature  zero,  so 
that   Brii  —  hu=^0   when   a-  =  0  ,    can  be  obtained  by  the  aid  of  Art.  53. 


Chap.  IV.]  TEMPERATURE  OF  AIR  ZERO.  125 

Let    u  =  Ui  +  »2  where  u-^  is  the  temperature  that  woukl  be  due  to  the  given 
source  if  we  had  no  boundary  at  the  origin,  so  that 

?^i  =  — j=-  G-  ^-^.  [Art.  53  (2)] 

2a\7rt 

D^.u  —  Jilt  =  Dj.Ui  —  livi  -\-  D,.i(-2  —  hi(.2  =  0    when   x*  =  0  . 

Therefore  £>,  u.  —  h  u.  =  —  (Z/,.  u^  —  hu^)  (1) 

when   a'  =  0  . 


But  ,    _ 


sfrrf 


when   X  =  0. 

This  is  easily  seen  to  be  the  vahie  to  which 


reduces  when   a'  =  0  ,    and  this  last  expression  is 

2aS/7rt 
and  therefore  satisfies  the  equation 

D,u  =  a^D^-tc;  (2) 

Q  (\  +  X)2 

since     -    j=.  e     ia-.t        is  the  temperature  due  to  a  source  at     x  =  —  X. 
If,  then,  we  determine  u^  from  the  condition  that 

T^      .  7  X  Q  /X-\-  X  ,  \  (k  +  xfl 

!>,(,.  _/„,,)  =  --i^^(^- 7.),-^  (3) 

taking  care  not  to  introduce  any  arbitraiy  constant  or  arbitrary  function  of  t 
in  our  integration,  lu  will  satisfy  equation  (2)  and  condition  (1). 

Integrating  (3)  [v.  Int.  Cal.  §  4,  page  314]  and  determining  the  constants  of 
integration  suitably  we  get 

Q       r      (A  +  xy       „,    .     /*      .       (A  +  xyi       -1 

"'  ^  ^^t  L^""^-  2^^^"y  ^-'-^^  dx^ .  (4) 

Therefore  the  solution  of  our  problem  is 

n  =  -^^[e-^^-{-  e-'-^-  -  2he'-  f.--"^^^  dxl  .  (5) 

2aV7r^L  J  J 


126  SOLUTION    OF    PROBLEMS    IN    PHYSICS.  [Art.  7L 

If  we  replace  Q  by  f(X)dX  and  integrate  from  0  to  co  we  get  as  the  solution 
for  the  case  where  u=f(x)  when  ;;  =  0  and  a- >  0 ,  and  Dj.u  —  hu^=0 
when   aj  =  0 


u  =  ^^^Cf(\)dX  p-  ^^  +  e-  '-iSf^  -  2he"-Je~ "—'-^  dx~] 


(6) 


For  an  interpretation  of  this  result  by  the  theory  of  Images  and  tlie 
extension  of  the  method  to  the  conduction  of  heat  in  n  dimensions  see  G.  H. 
Bryan,  Proc.  Lond.  Math.  Soc,  Vol.  XXII. 

EXAMPLE. 

Show  that  if  u=f(x)  when  t  =  0  and  D^u  +  h[F(t)  —  u]  =  0  when 
.!■  =  0  we  must  take  w  equal  to  the  sum  of  the  second  members  of  (6)  Art.  70 
and  of  (4)  Art.  69. 

71.  As  another  problem  requiring  a  slight  extension  of  Fourier's  Theorem 
let  us  consider  the  vibration  of  a  rectangular  stretched  elastic  membrane 
fastened  at  the  edges,  that  is  of  a  rectangular  drumhead. 

If  two  of  the  sides  are  taken  as  axes  and  the  plane  of  equilibrium  of  the 
membrane  as  the  plane  of  XY  the  equation  for  the  motion  of  the  membrane  is 

r^z  =  c%i)^z  +  i>,fz)  (1) 

see  [x]  Art.  1. 

Let  the  membrane  be  distorted  at  the  start  into  some  given  form  ,-  =f(oc,  y) 
and  then  allowed  to  swing.     Our  equations  of  conditions  are  then 

(2) 
(3) 
(4) 
(5) 
(6) 
(7) 
We  can  get  a  particular  solution  of  (1)  by  our  usual  device.    Assume 


and  substitute  in  (1)  .  We  get  y-  =  e-(a"-^  +  /3^)  as  the  only  relation  that 
need  hold  between  a,  yS,  and  y,  in  order  that  ;s  =:r  e  »-^  +  ^^  +  v'  may  be  a 
solution.     This  gives 

Therefore  ^  _  g  a.  +  ^^  ±  c^  V^IT^ 

is  a  solution  of  (1)  no  matter  what  values  are  given  to  a  and  /3. 


,t  =  0 

when 

x  =  i) 

z  =  0 

X  =  a 

z  =  0 

y  =  o 

z  =  0 

y  =  b 

z=f{x, 

.2/)" 

t  =  0 

,.v  =  0 

t  =  0 

Chap.  IV.] 


VIBRATION    OF    A    KECTAKGULAK    DKUMHEAD. 


l^-^ 


Replace  a  cind  (3  by  at  and  /3i  and  we  have 
as  a  solution,  and  from  this  we  get 


z  =  sin  (a.r  +  |8.v  ±  r'f  Va'^  +  /3^) 


and 


cos  (aa-  +  j8y  ±  '-'^  Va-  +  /S") 


(8) 
.(9) 


as  particular  solutions  of  (1),  a  and  /8  being  unrestricted. 

From  (8)  and  (9)  we  can  get  solutions  of  the  foUov/ing  forms 


=  sin  ax  sin  ygy  sin  cf  \a-^  +  (i^ 


=  sin  ax  sin  y8^  cos  ct  \a^  +  ^- 


=  sin  ax  cos  /Sy  sin  ct  \a'^  -\-  (3'^ 


sin  ax  cos  /3^  cos  c;^  \a^  +  yS* 


s;  =  cos  ax  sin  ;8y  sin  ct  \a^  -\-  ^" 
z  =  cos  ax  sin  /?^  cos  ct  ^a^  +  ^- 
s:  =  COS  aa;  cos  /Si/  sin  c^^  Va-  +  yS" 


(10) 


s;  =  COS  ax  cos  /3//  cos  e?^  Va"  +  /3"^ 

each  of  which  will  satisfy  equation  (1).     The  second  of  these  will  satisfy  also 

(2),  (4)  and  (7)  whatever  values  be  taken  for  a  and  ft.     It  will  satisfy  (3)  and 

7ii7r  mr 

(5)  if  a  and  ft  are  equal    and   —   respectively. 

If,  then,  we  can  so  combine  terms  of  the  form 


.     imrx    .     mrii 

sm •  sm  —r-  cos  cirt 

a  h 


V: 


as  to  satisfy  (6)  our  problem  will  be  completely  solved. 

This  can  be  done  if  we  can  express  f(x,  y)  as  a  sum  of  terms  of  the  form 

^  sin sin— —=-,     the    sum    and    the    function    being  equal  when   .'■   lies 

between  0  and  ct  and  y  between  0  and  h. 

f(x,  y)  can  be  expressed  in  terms  of    sin  — 

regard  y  as  constant.     We  have 


by  Fourier's  Theorem  ii    \\  - 
A^,  y)  =  %  '^m  sill  "^  (11) 


128  SOLUTION    OF    PEOBLEMS    IN    PHYSICS.  [Akt.  71. 

where  «m  =  -  ff(^^  V)  sin  ?^  dk  .  (12) 

f(X,  y)  in  (12)  is  a  function  of  y  and  may  be  developed  by  Fourier's  Theorem. 
We  have  f{\, y)=^ b„  sin  ^  (13) 

h 

where  K  =  j  ff(>^,  /*)  sin  '-—^  dfx, .  (14) 

Substituting  for  /(A,  y)  in  (12)  the  value  just  obtained  we  have 
««  =  -J  a(J  ^^^J  /(^'  f")  sm  ^^  sm  -^  df^j  sm  -^ 

m=l    0  0 

and 


Hence  '^  =  XX  ( ^"''^  ^^^  ~V  ^^^  "^  "^^^  ^'^'^  \  -^T  +  ^ ) '  ^^''^ 

n!=lH=l 

a  h 

where  ^,„_„  =  —  |  f/A  | /(A,  /i)  sin  - —  sin  ——-  dfx  .  (17) 

0  0 

is  our  required  solution. 

EXAMPLES. 

1.  Show  that  if  the  membrane  starts  from  its  position  of  equilibrium  but 
Avith  a  given  initial  velocity  impressed  upon  each  point  so  that  z^O  when 
/»  =  0    and   DfZ  =  F(x,y)    when    f  =  0    the  solution  is 

z  =  —y\  y,{A,„  „      ,  sm sm  —r^  sm  cirt  J!!L^L\ 


+ 


b^ 


-here 


^-" = ^/'^/^(^'  ^>  ^^^^  "^ ''''  T  '''^ 


Chap.  IV.]  RECTANGULAR    DRUMHEAD.  129 

2.    If  there  is  both  initial  distortion  and  initial  velocity 


^  —  -jZj    Zj  Sill sm  -—    A,„  „  cos  rirt  V  —  +  jr  +  ^',n  »  sm  <?7r^  a  —  +  77 

ab '^    ^^  a  ^     L  ''"        ''"  "'       '^"J 

m  =  1  n  =  1 

n  b 

where  ^,„_„  =  j  c^A  |  F(\,ix)  sin  — —  sin  — —  rlfx  , 


and  7),„  „  = ^   i  r/X  j  i^(A,  /x)  sin  ^^-^^'  sin  -^  r7/A . 


- 
a- 


+ 


7*- 


3.  Obtain  a  particnlar  solution  of  (1)  Art.  71  by  assuming  -j  =  j^'.A'.  I'. 
where  T  is  a  function  of  t  alone,  A'  of  x  alone,  and  I'  of  y  alone. 

72.  A  number  of  interesting  conclusions  can  be  drawn  from  the  results  of 
Art.  71  and  Exs.  1  and  2. 

(a)  No  one  of  the  three  values  of  z  is  in  general  a  periodic  function  of  t, 
and  consequently  a  vibrating  rectangular  membrane  will  not  in  general  give  a 
musical  note. 

(li)     A  stretched  rectangular  membrane  can  be  made  to  give  a  musical  note 

by  starting  the  vibration  properly.     For  if  the  initial  circumstances  are  such 

that  the  solution  reduces  to  a  single  term,  as  will  be  the  case  if  the  initial  dis- 

Hiirx         n^it 
tortion  in  the  problem  of  Art.  71  be  such  that    f{x,  y)  =  .4„^  „  sin sin  — — - , 

or  the   initial  velocity  in  Ex.  1  be  such  that     F(x,  y)  =  B,,, ,,  sin '-  sin  — -  , 

^  '  a  h 

or  the  initial  distortion  and  initial  velocity  in  Ex.  2  be  the  values  just  given, 
then  the  vibration  will  be  periodic  and  will  have  the  period 

o 

(1) 


Since  T  is  a  function  of  m  and  n  and  m  and  n  are  any  whole  numbers,  the 
same  membrane  is  capable  of  giving  a  great  variety  of  musical  notes  of  differ- 
ent pitches.  If  m  and  n  are  both  unity  we  get  the  lowest  note  the  membrane 
can  give,  which  is  called  its  fundamental  note.     Its  period 

T  =  — £=.  =  — SL.  (2) 

If  m  and  w  are  both  equal  to  k  we  get 

T,=        ^^i_;  (3) 


130  SOLUTION    OF    PROBLEMS    IN    PHYSICS.  [Art.  73. 

therefore  the  iiiembraiie  can  be  made  to  give  any  harmonic  of  its  fundamental 
note. 

More  than  this,  since  as  we  have  seen 

9 

T      — 


is  the  period  of  any  note  tlie  membrane  can  give,  and  since  if  in  and  n  are 
replaced  by  ink  and  iik  we  get 

•^  mk,  nk-  ^^ 


ek  \—  +  - 


the  membrane  can  sonnd  all  the  harmonies  of  any  note  which  it  can  give. 

(c)     In  the  case  considered  above,  where  the  solution  reduces  to  the  single 
term 


.     iii'Trx    .     mrii 

z  =  sm sm  ^r^     A,. 

a              0     \_ 

iir 
, ,,  cos  cTrt  \  — T  H 
^  (/■■ 

a            2a            ?><( 
if    x  =  -  ,   or   — ,   or   —  •  • 
III             III              III 

(hi  —  1)>I 

■    or 

III 

a               2a 
the  lines    .r  =  —  ,   .<■  =  —  ,  • 
III              III 

z  =^  0     for  all  values   of  f,   and 

i-emain  at  rest  during  the  Avhole 
vibration  and  are  nodes.     The  same  thing  is  true  of  the  lines 

h  2h  ?>h  (II  —  l)l> 

y  =  -,    1jr=  —  ,,/  =  —  ,■■■   ,j= . 

a  II  n.  II 

73.  If  the  membrane  is  square  it  may  have  much  more  complicated  nodes 
than  if  the  length  and  breadth  are  unequal,  as  in  this  case  the  period  of  any 
term  of  the  general  solution  reduces  to 

o„ 

(1) 


and  there  will  in  general  be  two  terms  having  the  same  period,  and  a  musical 
note  of  the  pitch  corresponding  to  that  period  may  be  produced  by  initial  cir- 
cumstances that  bring  in  both  terms.     Thus 

.     imrx    .     niri/r  ^'^^  ,/~^t -^   i    r>         ■     '"^'' »/~^rn :,"! 

z  ^=  sm sm  — -      J,„_„  cos  \iir  -[-  ir  -f-  /),„_„  sm  - — •  S/nr  -\-  ir 

CC  (X       \ it  Ct  J 

,       .      IITTX     .      mTTI/  r    .  CTTt     / — 3— .^    ,      „  .       ^'TT^     /^ r,~l 

+  sm sill ~     A„  „,  cos  ■ —  \iir  -j-  ir  +  J^,,  „.  sm  —  Snr  -\-  ir 

a  a     L  «  «  J 


Chap.  IV.]  NODES    OF    A    SQUARE    DEUMHEAD.  181 

is  a  form  of  vibration  that  will  give  a  musical  note.     Let  us  write  this 

'"^^  J~~^^ -iV   A      ■      '^'''^■''     ■       """'/     1      7,     ■      ^'''■•^'     •       W'TTV"! 

z  =  cos   —  \ III-  -]-  i)^\  A  sm sm  — -  -\-  B  sm sin 

'I  L  ((  a  a  a     J 

I      •     '^"^^  J~^r~\ — ?r  ^i    ■     ^Ji'^-x^    ■     nirii    .    ^^    .     717TX    .     i)i7ri/~\ 

+  sm V"'   +  n       C  sm sm  — '-  +  D  sm •  sm (2) 

(I  L  ('  ((  11  'I    A 

and  in  studying  the  forms  of  musical  vibration  of  which  the  membrane  is 
capable  we  may  take  A,  B,  C,  and  D  at  pleasure.  Consider  the  simple  case 
where  ^4  =  C  and  B  =  D;  then  (2)  reduces  to 

/  ,    .     7mrx    .     mry    .     ^,    .     nirx    .     miriiX/        cirt    ,— r-; , 

z^\A  sm •  sm y  B  sin sm )|  cos  —  Vw   +  ;r 

\  a  a  a  a    f\         a 

,       .      CTTf     I p- \ 

+  sm  —  S  III-  +  II- J .   (3) 

Values  of  X  and  y  that  will  reduce  the  first  parenthesis  in  (3)  to  zero  will  cor- 
respond to  points  of  the  membrane  remaining  motionless  during  the  vibration. 

Let  us  consider  a  few  cases  at  length. 

(ft)     If   VI  =  1    and    n^l ,    the  first  parenthesis  in  (3)  becomes 

(A  +  B)  sin  —  sin  '^  , 

^  a  a 

which   is   equal   to  zero   only  Avhen     x  =  i)     or     y^O,    or    .r  =  <■/    or    ?/=»'/, 
that  is.  for  the  four  edges  of  the  membrane.      If,  then,  the  membrane  is  sound- 
ing its  fundamental  note  it  has  no  nodes. 
{I))      If    ni  =  1    and    /^  =  2 ,    we  have 

.     TTX    .     Iirij  .     2'Trx    .     irii 

A  sm  —  sm +  P>  sm sm  ^  =  0 

a  (I  a  a 

to  give  the  nodes. 

TTX  27r//  ri 

Let    i?  =  0,    then    sin  —  sin — ^  =  0,     which  is   satisfied  by    >/=-:    and 
ft  a       ^^  -^    ''       2 

in  addition  to  the  edges  the  line    y  = -,    is  at  rest  and  is  a  node. 

If   J  =  0    a-  =  -    is  a  node. 

li   A  =  B  ^ 


2'Try   .     .     2'jrx    .     iri/ 
— -  +  sm sm  — ^  =  0 


TTX     .       Try  TV     ,     o     •       '"■•^'  TT'^'     •      TT'/         /^ 

2  sm  —  sm  -^  cos  -^  +  2  sm  —  cos  —  sm  — ^  =  0 
ft  a  a  a  a  a 

•     ■"■//  /        "T^//    1  '^^\ 

nn  — ^  (  cos  -^  +  cos  —  I  =^  0 . 
ft   \         ft  ft  / 


.      TTX 

sm  —  sm 


132 


SOLUTION    OF    PEOBLEMS    IN    THYSICS. 


[Art.  73. 


The  first  factor  gives  the  four  edges  of  the  membrane.     The  second  Avritten 
equal  to  zero  gives 


'Try  TTX  /  ■7r.i\ 

—  =  —  cos  —  =  cos  I  vr I 

a  a  V  a  / 


TTIj TTX 

a  a 


;r  +  ^  =  a, 


which  is  a  diagonal  of  the  square. 
If  B  =  -A 

.     TTX    .     27n/ 
sm  —  sm  — - 
a  a 


sm sm  — ^  =  0 


which  is  the  other  diagonal  of  the  square. 

Other  relations  between  A  and  B  will  give  Trigonometric  curves  of  the  form 

Try  B         TTX 

cos  ^^  ^ -.  COS  — 

(I  A  a 

which  are  easily  constructed  and  Avhich  obviously  all  agree  in  passing  through 
the  middle  point  of  the  square. 

We  give  the  figures  for  a  few  of  the  cases 


Chap.  IV.] 


NODES    OF    A    SQUARE   DRUMHEAD. 


133 


(c)     If   m  =  11  =  2    we  have 


(J  +  B)  sm sm  — ^  =  0 


to  give  the  nodes,  which  are  merely  the  lines 

a  a 

X  —  -  ,    and    11  =  7,- 


This  form  gives  the  octave  of  the  fundamental  note. 
{d')     If    la  =  1    and    u  =  3    Ave  have 


,      .       irX      .       StTI/     .      „      .       OTTX      .       TTII 

A  sm  —  sm  — =-  +  B  sm  — —  sm  — ^ 

a  (I,  a  a 


to  give  the  nodes. 
If   A  =  0    we  get 

If   B  =  0    Ave  get 
If   A  =  —  B   Ave  get 


a         ,  2a 

=  -    and   x=:—- 

o  o 


//  =  3    and   y  =  --  . 


.       TTX     .      StTI/  .       OTTX     .       TT?/ 

sm  —  sm  — '■ sm sm  — ^  =  0 


(1) 

(2) 


rx  .  77)/F 
—  sm  — ^  ^ 
a  a   \_ 


a  J 


TT//  irx       ,^ 

cos^  — cos^  —  =:  0 

a  (I 


(  TT//  '^^\(  Try     ,  TT.rX 

(  COS  -^ COS  I  (  COS  -^  +  COS  — ■ )  =  0 

V         (/  a  J  \         a  a  / 


If   A=z  B   we  get 


.? —  //  =^  0    and   ./•  -\-  ij=i  a  . 


TT//                     TT.r         1 
cos-^ 1-  COS^ ^  - 


(3) 


COS  '-  -\-  COS  =  —  1  , 


(4) 


a  Trigonometric  curve  easily  constructed. 

For  other  relations  between  A  and  B  we  get  more  complicated  Trigonometric 
curves  coming  under  the  general  form 


A  cos  — -  +  B  cos =  — 


A  +  B 


(5) 


134  SOLUTION    OF    PROBLEMS    IN    PHYSICS. 

which  all  agree  iu  containing  the  points 

(a   (i\    (a    2a\    /2a   a\  ,     /2a.   2a\ 


MISCELLANEOUS    PROBLEMS. 

I.     Lofjarithmic  Potential.     Polar   Coordinates. 
1.    Show  that    D^V-{-  D;jV=()    becomes 

if  we  transform  to  Polar  Coordinates. 


2.    If  in 
we  let   V^  R.<^   we  get 

^^  A  cos  a(^-\-  B  sin  a<^ 

whence 


(1) 


^  =  A^  cos  (a  log  /■)  +  Bi  sin  (a  log  r) 


V^  —  cos  a^ 


r= —  sin  a< 


V=  e"-'^  cos  (a  log  r) 
J'=  e«'f>  sin  (a  log  /•) 

r=  e-"*  cos  (a  log  r) 
r=  e-""^  sin  (a  log  ?■) 


J^=  cosh  ac^  cos  (a  log  ;•) 
Y  =  cosh  a</)  sin  (a  log  ?•) 

P^=  sinh  a<^  cos  (a  log  ?•) 
V-=  sinh  a<^  sin  (a  log  r) 


are  particular  solutions  of  (1). 

3.    Show  that  if  F  satisfies  (1)  Ex.  2  and    V=f{4>)    when   r  =  a 

V=  -  l\,  +  X(~)"V^«  *^°^  ""^  +  -''»  ^^^^  "'^)    ^^^'    ''  *^  "■ 

ni=l 

and  ^— T7^'o  +  X(~)   (^'m  cos  7«<^  +  a,„  sin  ?m^)    for    r>a, 

where  ^>^  =  -  Cf(4>)  cos  7n(f>.d(f)    and    o„j  =  -  r/'(<;f))  sin  ??;^.(/<^ 


136  MISCELLANEOUS    PROBLEMS. 

4.    Show  that  if  F  satisfies  (1)  Ex.  2  and    V—f(r)   when    ^  =  0    and    r>0 


V  = 


^  ^  cosh  -  (A  -  log  r) 

=  -  sin  ^  I  /-(eM  — —^ d\ 

TT        2  J  •  ^    ^  cosh  (X  —  log  /•)  —  cos  <t> 


1/^/./   >N  7.   /^cosh  a(7r  —  d)) 

1    .    <!> 

5.  If    F=l    when    (^  =  0    and   0  <;■<!,    and    V=0   when    <^  =  0    and 
/■>1 

(  sm^         )  2v//-.sin^   -" 

6.  If    r=/(r)    when    cf>  =  0    and    r=0    when    cf>  =  (3 

V=-  ff(e^)clX  n^^^\(l^-J^  cos  a(A  -  log  .)•</« 
ttJ  "^  ^  J        smh  /3a 

^  A  sin  ^  r Zle^A 

2/3  fij  ,    TT  ^^  .  ^  TT        ' 

-oc  cosh  ^  (A  —  log  r)  —  cos  -  c/i 

if     0<<^<y8. 

7.  If    F=0    when    <^  =  0   and    V—F()-)   when    ^  = /3 

r=^/.(.,.A/^:cos.^^ 

-»  cosh  —  (A  —  log  /•)  +  cos  —  0 

8.  If  V=x(>-)    when  <^  =  0  and  r<a,     V=0    when  ^  =  (3,    and 
F  =  0  when  r  =  a 

0 

y—    1    „:„  '"■'^  r    /  ..x\  r  f''A 


-_3in^/,(„.,r 


cosh  ^  (  A  —  log  -  )  —  COS  — ^ 


cosh  -  ( A  +  log  -  )  —  COS  -^' 
13  \  a/  (3 


LOGARITHMIC    POTENTIAL.  137 

9.  If    7^--^0  when    r  =  l,    V—1    when    <^  =  0 ,     V=0    when    <l>  =  -^ 

2  n  —  r-  ~I 

r=  -  tan-'      ,-— — :,  ctn  </>      • 
TT  Li  +  r-  J 

TT 

10.  If    V—0    when    r  =  1  ,    F=  1    when     </>  =  0  .    T'=  1    when    <t>  =  tj 


bau-i      — r-^ — —      • 
\_2r-  sm  2(i  J 


TT  L2r-  sm  J(^. 


11.  If    V=f(<t>)    when    r=^^,     r=0    when     <^  =  0,    and    V=0    when 

7  =  2^a,,,  l-J  n  sm  — — -      i±     r  >  <: 
where  «,„  =  -  Cf\<i>)  sin  ^^^  (/<^    and    0  <  </>  <  ;8. 

0 

12.  If    V=^ /(<!>)    when    >■  =  «.,    F=  0   when   r  =  h,    r'=0   when   </)  =  0, 
and    V=  0    when    <^  =  /3 ,    then  if   a  <>'</>    and    0  <  (/>  <  ^ 

m=l        rt     p     —  b     P 

where  a„,  =  -J  /((^)  sm  — -  f/<^  . 

13.  If    V^F((I>)    when    )'  =  b,    r=()   when   r  =  a,    V^O  when    (^  =  0, 
and    V=0    when    <^  =  i8,    then  if   a  <  >•  <  ^>   and    0<  <^  <  ^ 


^=X{ ^ — =  lU"- 17.) '  J "» ^■" -;8-  \ 


3 

where  «„j  ~  ^  I  ^(^)  '''^^^  — o~  '^'^  ■ 

P»y  P 

14.    If    V=xQ')    ^''l^en    <^  =  0,    F=0    when    c/)=r^,    r=0  when    r=a, 
and    r=  0    when    /•  =  ^»,    then  if    a<y<b   and   0  <  <^  <  ^ 

»<  =  cc  ^-   1     W7r(/3  —  <^) 

T^ x^  (        "         lotr  ^>  —  loar  a    .     mirilo^  r  —  log  a)  '} 

'^  —  Zji  «m ^^ ^--  sm  — T^ — —^  \ 

^  i.  .    ,  mir  log  b  —  log  a       ) 


log  Zy  —  log  a 


138  ]VnSCELLANEOUS    PKOBLEMS. 

log- 

where  «,„  =  . 7 -, |  x(^'<^  )  sm , •  d.r . 

log  />  —  log  a  J  '^^       '        log  h  —  log  a 

15.    If    V=if/()-)    when    <^  =  y8,    7^=0    when    <^  =  0.    7'=0    when    ;-=:r/, 
and    F=  0    when    r  =  b,    then  if   a  <)-<h    and   0  <  <i>  <  /3 


r=X 


snih 


^  log  i  —  log  a    .      «i7r(log  > —  log  a)  | 


sm 


1     '"     .    ,  VITT  log  /y  —  log  a         S 

smh 


log  />  —  log  a 

log 
2  /*  DITTOC 

where  a™  = :; : ^ I  ib(ae^)  sin , ; ^ dx. 

log/>  —  log  r/J  ^^  log/>  — logrf 

II.     Potential  Function  in  Space. 

1.  Show  that 

/(■-^,  y)  =  -\  fda  Cd(3  Cdk  Cf(\.  fi)  cos  a  (A  —  .r)  cos  ^(/x  —  j/).d/x, 

for  all  values  of  x  and  //. 

2.  Find  particular  solutions  of   D^V-\- D,JV-{- D^J^^=0    in  the  forms 

T^=  e  *  .-^a=  +  P^  cos  (ax  ±  jBi/) 
r=  e  *  =  ''«' +  ^"-  sin  (ax  ±  (Sij) 


V=  sinh  z  SI  a-  +  p-.  sin  (a.r  ±  /?//) 


r=  cosh  s:  Va'  +  /?-.  sin  (aa*  ±  y8//) 

&c. 

3.    Given    D;V-\-  D^'^ -\-  D;V=0 ,  and    V=f{x,y)  when  ,t':=0,  solve  for 
positive  values  of  z. 


Remit. 


1  Cdx  C  ^/"(^^z^)^^  . 


4.    Confirm  the  result  of  the  last  example  by  showing  that  if  f(x,  y)  is  inde- 
pendent of  y 

Y^  1  r  zf{\,ti)dX  .,.  Ex.  3  Art.  45). 

irjz^+(k-xy  ^ 


POTENTIAL    FUNCTION    IN    SPACE.  139 

5.  If  D^  F  +  />,;  r  +  I);  / '  =  0 ,  and  V  =  1  when  ,~  =  0  for  all  points 
within  the  rectangle  bounded  by  the  lines  a- =  «,  x  =  —  a,  >j^=b,  and 
yz=  —  h;  and  F=0  when  ,v  =  0  for  all  points  outside  of  this  rectangle, 
then 

.,     ,._      b-U      f  TT       1  (a  -  xy\h  -  yf  -  z^j,,  -  aif  +  (b  -  yf  +  z^^ 

-'^  ''  -  s/(^r^^'^  1  2  "^  2  ''^     {a  -  xy(h  -  yf  +  ,.^[(«  -  x)'  -^  (h  ~  y^  +  z^-] 

1    ._,  (a  +  -rfih  -  yy-  -  z^a  +  .-)^  +  (^  -  !/Y  +  .-^] 

-t-  2  (,,  +  ,rf(J^  -  yy  +  .^[(«  +  ;,.)2  +  (^,  _  y)2  +  .V^] 

.   _1±J^  t  TT       1  0^,  -  ■t)'^(^>  +  y)-^  -  z\{a  -  a-y  +  (b  +  yy  +  zT^ 

^  ^j:fyy  12-^2'  (a  -  xy(b  +  yy  +  zX(a  -  xy  +(b  +  yy  +  z^] 

,    1    .,  {a  +  ^ryU,  +  yy  -  .^[(.  +  .r)^  +  (^  +  Z/)^  +  ^^]  1 

^  2  ■       (a  +  .'■)^(^  +  uT  +  ^TC'  +  .'f  -^{b  +  yy  +  ^^]  ^ 

if    —  ^;  <  X  <  rt,    and 

■iTT  >-  —  ^=  "1       sin -^-7 ^-j — T— — , — — — ; — — 

yj(b  -  yy  I  («  -  -^yi^  -  II)-  +  ■"[(«  -  ''•;-  +  i!>  -  yf  +  -'] 

-  .in->  («  +  ^y{b  -  yy  -  ^{a  +  ^0-^  +  {b  -  yy  +  ^^] 

(« + '^n^  -  yy + ^T(« + -^o' + c^'  -  yy + •^'] 

'^iw+w  '^      (''  -  ''y^^' + •'^)' + '^'[(^  -  '-y' + (^' + '^y + '^'^ 

(..  +  xy^b  +  7/)^  +  .-T(«  +  '^O"^  +  (^  +  y)-^  +  ^^jr 

if    a;  <  —  «    or    x  >  «  . 

6.  If  the  value  of  the  jDotential  function  V  is  given  at  every  point  of  the  base 
of  an  infinite  rectangular  prism  and  if  the  sides  of  the  prism  are  at  potential 
zero  the  value  of  V  at  any  point  within  the  prism  is 

Tr       "i  -^    X-*'      „-  \J^^"-  +  "^    •     m'rrx    .     miry  r  .^  f,^^^      ,     .     mirX    .     mr/x  , 

,« = 1 ,( = 1  0    0 

If    V=l    on  the  base  of  the  prism  this  reduces  to 

.     (2m  +  l)7ra;    .     (2n  +  l)7r?/ 
m=  ..  „=  X sin  ^ '- —  sm  ^^ 

TT-'  ^    ^  "'  "-  (2w^  +  1)  (2»  +  1) 

»i  =  0    n  =  0  ^ 

7.  If  the  value  of  the  potential  function  on  five  faces  of  a  rectangular 
parallelopiped,  whose  length,  breadth,  and  height  are  a,  b,  and  c,  is  zero,  and 


140  MISCELLANEOUS    PROBLEMS. 

if  the  value  of  V  is  given  for  every  point  of  the  •  sixth  face,  then  for  anj- 
point  within  the  parallelopiped 

smh  7r(c  —  z)  \—  +  - 
V=  >     \,Ann —  sm sm  — - 


where  J,„ „  ^—  \dk  \f{\,  y)  sin  -^- —  sin  —j^  ^A*- 

8.  If  the  value  of  the  potential  function  is  given  on  two  opposite  faces  of  a 
rectangular  parallelopiped  and  is  zero  on  the  four  remaining  faces,  then  within 
the  parallelopiped 


n^^n^  sinh7r(c-,.)^-+- 


.     viTTX    .     ntni 
sm sm  -    ■ 


.    ,  \m-   ,    n-  (^  b 

smh  7rc\\^  -\-  -- 

^   ri-  I- 


^here 


and 


m  =  :r.n  =  ^  Smh  'TTZ  V—  +  — 

Xx-v  T^  *  «         '>     .     niTTX    .     nirii 

S  ""'•'  ~, — W^' '"  ^  ""  ^ 

m=  1  H=  1  smh  ire  \  —  -\-  — 

'  a-       Ir 

a  b 

An,n  =  ^  J  ((^  I  f(K  HO  sm  —^  sm  —^  rf/^ 
^w,H  =  —,  I  ^^-^  I  ^(^j  h)  sm sm  —-^  dfi . 


9.  If  the  value  of  the  potential  function  is  given  at  every  point  on  the  surface 
of  a  rectangular  parallelopiped,  what  is  its  value  at  any  point  within  the 
parallelopiped? 

III.      Conduction  of  Heat  in  a  Plane. 

1.  Find  particular  solutions  of    I)fU  =  a-(I)^u -\- I),ju)    of  the  forms 

u  =  e- "'("'  +  ^"->'  sin  (ax  ±  j3i/) 
u  =  e~  "'''"'  "^  ^'^'  cos  (ax  ±  (Si/) . 

2.  Given  the  initial  temperature  of  every  point  in  a  thin  plane  plate,  find 
the  temperature  of  any  point  at  any  time. 


CONDUCTION    OF    HEAT    IN    A    PLANE.  141 

3.    For  au  instantaneous  source  of  strength  4^  at  (A,  /a) 

'^  =  -. r  e  4n2,  V.  Art.  5-1 

47ra-f 

For  an  instantaneous  dovMct  of  strength  P  at  (0,  /a)  with   its   axis   perpen- 
dicular to  the  axis  of  Y 

PX  .r^+(M-.v)^  .      ^      ^^ 

f         ia2t  V.  Art.  54. 


87ra4^2 


For  a  permanent  doublet  of  strength  P  at  (0,  /x)  with  its  axis  perpendicular 
to  the  axis  of  Y 

—  JL  --^  _.d+Oi^iO! 

If  the  strength  of  the  doublet  were  Pdjx  and  the  heat  were  uniformly 
generated  and  absorbed  along  the  element  dfx  of  the  axis  of  Y  beginning  at 
(0,  ix)  we  should  have 

P         _  a-2  +  (|^-y)2  xdjJL  P  ^^(,u.-vf     ,  ^  ,  t^—  If 

«  =  7) — "2  ^        4a2«         o  I   , To  =  ri — -o  e        i^^s;^       f?  tan-' , 

and  since    d  tan"'  — ^    is  the  angle  ARA\  where  A  and  A'  are  the  points  (0,  (x) 

and     (0, /u,  +  r/yu.)     and    P.    is    the    point    (■*',//),     ?^  ==  0     when    .ri=  0     unless 

P 

fjL  <.  1/  <  fi  -{-  dfi,     in  which  case     «  =  ^^Tj     if  x    approaches    zero    from   the 

positive   side ;  and    u  =  0    when    t  =  0    except   in  the  element  d/x.      If  then 

ti^O    when    t  =  0    and    n  =f(y)    when    x  =  0    we  have  only  to   suppose   a 

doublet   of   strength  2a\f(x)dx  placed  in  each  element  of  the  axis  of  Y  and 
then  to  integrate;  we  get 

ttJ  X- -\- (/x —>/)- 

For  a  permanent  doublet  of  strength  F(f)  at  (0,  /x)  we  have 

II 

^   1    r     xF(0)         ^2+(^_,,^2     A     xF'(t)         .^+(^-,f-    -| 


142  MISCELLANEOUS    PROBLEMS. 

From  the  reasoning  above  this  must  be  zero  when  t=^0  except  at  the  point 
(0,  fi),  must  be  2a^F(t)  at  the  point  (0,  fi),  and  0  at  every  other  point  of  the 
axis  of  Y  when  t  is  not  zero. 

Hence  if   m  =  0    when    t  =  0    and    u  =  F(i/,  t)    Avhen    ./•  =  0 

ttJ  :,-  +  (/*  -  >/)-  "^       ttJ     O  ./■-  +  (/i  -  i/Y         ""  ^'    '' 

For  an  extension  of  this  solution  by  the  method  of  images  to  the  case  where 
there  are  other  rectilinear  boundaries  and  for  its  application  to  the  correspond- 
ing problems  in  the  flow  of  heat  in  three  dimensions  see  E.  W.  Hobson  in  Vol. 
XIX  Proc.  Lond.  Math.  Soc. 

4.  If  the  perimeter  of  a  thin  plane  rectangular  plate  is  kept  at  the  tem- 
perature zero  and  the  initial  temperatures  of  all  points  of  the  plate  are  given, 
then  for  any  point  of  the  plate 

n  =  j-^2^   2_i<i~''     \b^^  ^r  sm  -y-  sm  —^\  d\\  f(X,  fi)  sm  -^  sm  -^  dfi . 

if  b  is  the  length  and  c  the  breadth  of  the  plate. 

5.  A  large  mass  of  iron  at  the  temperature  0°  contains  an  iron  core  in  the 
shape  of  a  long  prism  40  cm.  square.  The  core  is  removed  and  heated  to  the 
temperature  of  100°  throughout  and  then  replaced.  Find  the  temperature  of  a 
point  in  the  axis  of  the  core  fifteen  minutes  afterward.  Given  a:  =  .185  in 
C.G.S.  units.  Ans..  o2°.9. 

6.  If  the  prism  described  in  Ex.  5  after  being  heated  to  100°  has  its  lateral 
faces  kept  for  15  minutes  at  the  temperature  0°  find  the  temperature  of  a  point 
in  its  axis.  Ans.,  20°.8. 

IV.      Conduction  of  Heat  in  Sj^ace. 

1.  Show  that 

—  I  rfa  I  fZ/3  I  dy  I  d\  |  d/x  i  f(X,  fi,  v)  cos  a(\  —  x)  cos  y3(/x  —  y)  cos  y(v  —  z).dv 

0  0  0  _:x         -"i         -"S^ 

for  all  values  of  x,  y,  and  z. 

2.  Show  that 


fix, y,z)  =  2^  2^   2^  J,„.„^^  sm  —^  sm  — 


^^-•^=J;/'4^^^P^ 


where      A„^„„  =  ~  \  dX  \  d/x  {  f(X,  fx,  v)  sin  — ^  sin  —~-  sin  - —  dv 


for    0<. ,•<./.    0  <//</>.    0<,?<c. 


CONDUCTION    OF    HEAT    IN    SPACE.  143 

3.  Obtain  particnlar  solutions  of  I),nz:^a-(I);ii -\- l)~v -\- D;u)  of  tlie 
forms 

u  =  e- '''<''' +  ^"-  +  y'>'  sin  (ax  ±  (3>/  ±  y.^) 

u  =  e-a-(a^-  +  P'  +  y'>t  cos  (ax  ±  (Sy  ±  yz)  . 

4  Given  the  initial  temperature  of  every  point  in  an  infinite  homogeneous 
solid  find  the  temperature  of  any  point  at  any  time. 

=  -3  Ce-^'  d/S  fe-  y'  dy  Ce-  ^'/(x  +  2>>\ft.f^ ,  //  +  2a>Jt.y ,  z  +  2as[f.^)dB . 

5.  If  the  surface  of  a  rectangular  parallelopiped  is  kept  at  the  temperature 
zero  and  the  initial  temperatures  of  all  points  of  the  parallelopiped  are  given, 
then  for  any  point  of  the  parallelopiped 


U=2^2^     A-'^m,n,p'i    "'"'(t 


where       ^,„,„,p  =  r^  j  f/A  I  d^x  j /(A,  /w,  v)  sin  — — -  sin  ^^  sin-^ 


sm  -——  sm  — -  sm-'-— - 
bed 

plTV 


dv. 


6.  An  iron  cube  40  cm.  on  an  edge  is  heated  to  the  uniform  temperature  of 
100°  Centigrade  and  then  tightly  enclosed  in  a  large  iron  mass  which  is  at  the 
uniform  temperature  of  0°.  Find  the  temperature  of  the  centre  of  the  cube 
fifteen  minutes  afterwards.  An>i.,  '.\^°A. 

7.  An  iron  cube  40  cm.  on  an  edge  is  heated  to  the  uniform  temperature  of 
100°  and  then  its  surface  is  kept  for  fifteen  minutes  at  the  temperature  0°. 
Required  the  temperature  of  its  centre.  Ans.,  9°. 5. 


CHAPTER   V. 


ZONAL    HARMONICS. 


74.     In  Art.  16  we  obtained 

z  =  Ap^(x)  +  Bq^(a-)  (1) 

[v.  (6)  Art.  16]  as  the  general  solution  of  Legendre's  Equation 


(1  -  •^')  X7.  -  2^^  £  +  M'"'^ +  !)-  =  <> 


(2) 


dx'^  '  d,x 

m,  being  wholly  unrestricted  in  value  and  x  lying  between  —  1   and  1;  where 
^„(,,  =  1  _  "-C'^+l)  ,» +  ».0«-2)(m +!)(«  + 3)  ^. 

m(m  -  2)(,»  -  4)(«,  +  !)(<«  +  3)(»,.  +  5)    , 

3:    -t"  (^o; 


6! 


and 


.    .  _  ^.       (m  -  l)(m  +  2)    .,       (».  -  \){m  -  3)(m  +  2)(;»  +  4)     ,' 


and  we  found 


3!  '  5 

(m  -  l){m  -  3)(w  —  ^){m  +  2)(m  +  4)(/;^  +  6)    . 

7!  ^' 


V=  r'^q^,^  (COS  6) 


+  •••;   (4) 


(P) 


'J 

m  being  unrestricted  in  value,  as  particular  solutions  of  the  special  form 
assumed  by  Laplace's  Equation  in  spherical  coordinates  when  V  is  independ- 
ent of  ^;  that  is,  of  the  equation 


rD::{r  V)  +  ^^  D,  (sin  6  De  V)  =  0  . 


(6) 


*  Before  reading  this  chapter  the  student  is  advised  to  re-read  carefully  articles  9,  10,  13(r), 
15,  10.  and  18(r-). 


SURFACE    ZONAL    HARMONICS.  145 

For  the  important  case  where  m  is  a  positive  integer  we  fonnd 

.  =  .4P,,(:.)+iH>,„(.'')  (7) 

[v.  (10)  Art.  16]  as  tlie  general  solution  of  Legenclre's  Equation  (2),  whence 
F=r'"P,„(cos^)       ■ 

V=  -\-,  P,„  (cos  0) 

V  =>'"%)„,  (COS  0) 

V=  -^  Qm  (cos  6) 
are  particular  solutions  of  (6)  if  m.  is  a  positive  integer. 
p  i,.^  -  (2^^^-l)(2>M-3)--  X  V      _  JM^ 


(8; 


1) 


!(2m-l) 

m{in  —  1)  (rii  —  2)  (m  —  3)     ^^^ 
"^     2.4.(2m  —  1)  (2?/i  —  3)     ■^' 


•]      (9) 


[v.  (8)  Art.  16]  and  is  a  finite  sum  terminating  with  the  term  which  involves 
X  if  m  is  odd  and  with  the  term  involving  .r°  if  mi  is  even. 

It  is  called  a  Surface  Zonal  Harmnnic,  or  a  Legendre's  Coefficient,  or  more 
briefly  a  Legendrian. 

^™^' ^       (2>N-\-l)(2m-l)---lLx"'-'^  2.(2»;,  +  3)      x'"  +  ^ 

(>»+l)(m-^2)(m  +  3)(m  +  4)     1 
"^  2.4.(2///. +  3)(2;»  + 5)  a."'  +  5-^ 

if   .r<  — 1    or   a->l.      [v.  (9)  Art.  16.] 

It  is  called  a  Surface  Zonal  Harmonic  of  the  second,  kind. 


•]    (10) 


Q,n(:r)  =  (-!)  — 


.d  and 

U^)  =  (-i)' 


r(»^  + 1) 
2»[r(|  +  i)] 

^  2  2.4.6.  •  •  •  (///  — 


-2  P>n(^) 


1) 


3.5.7.  ••  •  /// 
[v.  (13)  Art.  16]  if  ///.  is  odd  and  —  1<  .r  <  1 . 

r(///  +  l) 


Pr»(.^) 


(11) 


K^')! 


<U^) 


,    ,,^2       2.4.6.  •••/// 
-(-^'^  1.3.5..  ■■(,„- I)  »•-("> 

[v.  (14)  Art.  16]  if  m  is  even  and     —  1  <  r  <  1  . 


(12) 


146  ZONAL    HARMONICS.  [Art.  75. 

In  most  of  the  work  that  immediately  iollows  we  shall  regard  x  in  P,Jx)  as 
equal  to  cos  ^  and  therefore  as  lying  between  —  1  and  1.* 

75.     In   Article   9   the   undetermined   coefficient    (%  of   x"^  in  Pn(x)   was 
arbitrarily  written  in  the  form  ~ ^"  , '-^ for  reasons  which  shall 

now  be  given. 

In  Articles  9  and  16    z  =  t',n{^)    "^^^^   obtained   as  a  particular   solution   of 
Legendre's  Equation 

^^  ~  ""'^  S  ~  ^"^  !l + "'  ^"^ + 1)  .^ = 0  (1) 

by  the  device  of  assuming  that  z  could  be  expressed  as  a  sum  or  a  series  of 
terms  of  the  form  a^x'^  and  then  determining  the  coefficients.  We  can,  how- 
ever, obtain  a  particular  solution  of  Legendre's  Equation  by  an  entirely  differ- 
ent method. 

The  potential   function  due  to  a  unit  of  mass  concentrated  at  a  given  point 
(^i>  Vi,  «i)  is 

V=  ,  ^  (2) 

N/(.r  -  x,f  +  (^  -  VxY  +  (.^  -  .n)^ 

and  this  must  be  a  particular  solution  of  Laplace's  Equation 

Dl  V  +  Z>;  V  +  D/  r  =  0 ,  (3) 

as  is  easily  verified  by  direct  substitution. 

If   we   transform    (2)    to   spherical    coordinates    using    the    formulas    of 
transformation 

X  =  /■  cos  B 

y:=r  sin  $  cos  <^ 

s  =  r  sin  6  sin  ^  we  get 

V=  ,  ^  (4) 

y/,.2  _  2/Ti[cos  e  cos  $1  +  sin  6  sin  0^  cos(<^  —  <^i)]  +  'f 

as  a  solution  of  Laplace's  Equation  in  Spherical  Coordinates 

rD';{r  V)  +  ^l~  A(sin  6  A  V)  +  ^^  Dl  V=  0     [xiii]  Art.  1. 

If  the  given  point  (xi,  y^,  z^,)  is  taken  on  the  axis  of  X,  as  it  must  be  that 
(4)  may  be  independent  of  <^,    ^i  =  0 ,    and 

V  =  —=2==  (5) 

V'"^  —  2/Ti  cos  9  -f  rf 

*  English  writers  on  Spherical  Harmonics  generally  use  n  in  place  of  x  for  cos  d.     We 
shall  follow  them,  however,  only  when  we  should  thereby  avoid  confusion. 


Chap.  V.]       PARTICULAR    SOLUTION    OF    LAPLACE's    EQUATION.  147 

is  a  solution  of 

rI);'(rV)  +  ^.^-  De(sm  9 D,  V)  =  0 .  (6) 


Equation  (5)  may  be  written 

..      1 


(J) 


(«) 


V 1  —  -.V  cos  6  +  .V-'     is  finite  and  continuous  for  all  values  real  or  complex  of 

It  is  double-valued  but  the  two  branches  of  the  function  are  distinct  except 

for  the  values  of  z  which  make   1  —  2,"  cos  6-\-z-  =  0   namely   z  =  cos  9  +  /  sin  9 

and    ,-.'  =  cos  9  —  /  sin  9,    both  of  which  have  the  modulus  unity  and  which  are' 

crifirt/l  values. 

,  is    finite    and    continuous    except    for    the    values    of 

V/I  -  2z  cos  9-\-z' 

z  =z  009.  9  —  i  sin  9  and  ,v  =  cos  ^  +  /  sin  ^  for  which  it  becomes  infinite;  it  is 
double-valued  but  has  as  critical  values  only  these  values  of  .-.-.  It  is  then 
holomorjyjdc  within  a  circle  described  Avith  the  origin  as  centre  and  the  radius 
unity,  and  can  be  developed  into  a  power  series  which  will  be  convergent  for 
all  values  of  z  having  moduli  less  than  one.  (Int.  Cal.  Arts.  207,  212,  214, 
220.) 

If  then    r  >  i\  — =  can  be  developed  into  a  convergent  series 

'^'"  cos^  +  ^; 


v-^ 


involving  whole  powers  of  — . 

Let    z^Pm  ~7i    ^^6  this  series,  ^„j,  of  course,  being  a  function  of  cos  9.     Then 

[v.  (7)]  is  a  solution  of  (6).     Substitute  this  value  of  Fin  (6)  and  we  get 

As  this  must  hold  whatever  the  value  of  r  provided  r  >  )\  the   coefficient   of 
each  power  of  r  must  be  zero,  and  hence  the  equation 


must  l)e  true. 


iA/^&"^' 


148  ZONAL    HARMONICS.  [Akt.  75. 

But  as  we  have  seen  in  Art.  9  the  substitution  of   .r  =  cos  0   in  (9)  reduces 
it  to 

^^  -  "'^  S™  -  -^'  ^ + "'^"' + 1)  ^-  ==  0' 

and  therefore  ^^J'„i 

is  a  solution  of  Legendre's  Equation  (1). 

If    /•  <  /'i     — ^     can  be  developed  into  a  convergent  series 

a/i  —  —  cos  ^  +  -, 
\  )\  r{ 

involving  whole  powers  of  —  • 

Xj.»i  ''i 

j9„j  —   be  this  series.     Tlien 

(v.  8)  is  a  solution  of  (6) ;  substituting  in  (6)  we  get 

whence  it  follows  as  before  that 

^  =  Pm 

is  a  solution  of  Legendre's  Equation. 

But  7_>,„  is  the  coefficient  of  the  mth  power  of  —   in  the  development  of 

(1  —  2  —  cos  6  -\ — -„  I    2  according  to  powers  of  —  ,  or  of  the  mth  ijower  of  —  in 
the  development   of     ( 1  —  2  -  cos  ^  +  -yj    2   according  to   powers  of    -^  ,    or 

more  briefly  it  is  the  coefficient  of  the  mth  power  of  z  in  the  development  of 
(1  —  2xz  +  z^)-  2    according  to  powers  of  z,  x  standing  for  cos  6. 

(1  -  2,r..  +  z^-  h  =  \l-  z(2x  -  z)y  h 

and  can  be  developed  by  the  Binomial  Theorem ;  the  coefficient  of  z^  is  easily  , 
picked  out  and  is 


(2m-l)(2m-3)---l  r  _  ,,^  _  m(m-l) 
ml  [_'         2  (2  III  —  1) 


+ 


2)0»-3)^.,„_,      . 


2.4.(2//; -1)(2;«- 3)  J 

But  this  is  precisely  -P,„(a").     [v.  Art.  74  (9)] 

Hence  P^ix)  is  equal  to  the  coefficient  of  the  mth  power  of  .-  in 
the  development  of  [1  —  2xz  +  «^]"  i  into  a  power  series,  the  modulus  of  z 
being  less  than  unity. 


Chap.  V.]  PROPERTIES    OF    SURFACE    ZONAL    HARMONICS.  149 

76.     If    x  =  l    P„,(,x)  =  l.     For  if    x  =  l     (l-2xz  +  z^)-h    reduces   to 
(1  —  2z -\- z^)- 2    that  is  to    (l—z)-^,    which  develops  into 

H-^  +  ^^+^3  +  ^^  +  ---, 

and  the  coefficient  of  each  power  of  z  is  unity.     Therefore 

^„.(i)  =  i-  a) 

We  have  seen  that  if  m  is  even  P„^(oc)  contains  only  even  powers  of  .r  and 
terminates  with  the  term  involving  cc°,  that  is  with  the  constant  term. 

The  value  of  this  constant  term  can  be  picked  out  from  the   formula   for 

P„^{x)  [v.  Art.  74  (9)].     It  is     (—  l)^    '  t' a  c^^'.^ 7      5     or  it  can  be  found  as 

follows:  —  It  is  clearly  the  value  P„iQic)  assumes  when   a"  =  0;    it  is,  then,  the 
coefficient  of  z"^  in  the  development  of    (1  +  .~-)~  2^ ;    but 

and  the  coefficient  of  «'",  vi  being  an  even  number,  is    ( —  1)2  -^-^- ^— ^  . 

^  2.4.6  •  •  •  ni 

If  m  is  odd  Pj,^(x)  contains  only  odd  powers  of  x  and  terminates  with  the 

term  involving  x  to  the  first  power.      The  coefficient  of    this  term  can  be 

picked  out  from  (9)  Art.  74  and   is     (—1)    ^  '        '-—;     or  it  can  be 

2.4.0.  ■  ■  ■  (i)i  —  Ij 

found  as  folloAvs :  —  It  is  clearly  the  value  assumed  by    — "'^'       when    x  =  0 . 


It    is,  then,  the  coefficient  of  5;'"  in   the   development  of  3  . 

(1  +  z  )v 

^         = ..      -  ..  I  3-^  ...      3-5-T  .,   , 
(l  +  «^)l      ■'      2"  ^2.4'''        2.4.6''  ^ 

m-I  3  5  7  •  •  ■  III 

and  the  coefficient  of  «'"  in  this  development  is     (—  1)   2  '  ' — 

III  being  an  odd  number. 

77.      To  recapitulate: 

„       ,       1.3.5  •  •  •  (2???.  —  1)  r  m(7n  —  1) 

m(m-l)(m-2)(;M-3) 
"^     2.4.(2?»  — l)(2w  — 3)     ■ 

m(iii  —  1)(7»  —  2)(//^  —  ?-,)(m  —  A)(m  —  5)     ,„_,;,         ' 
2.4.6.  (2m  —  l)(2m  —  3)  (2?;^— 5)  ''"''"    "  ^        . 


(1) 


(3) 


150  ZONAL    HARMONICS.  [Akt.  77. 

m  being  a  positive  integer,  is  a  Siirface  Zonal  Harmonic  or  Leyendrian  of  the 
mth  order.  It  is  a  finite  sum  terminating  with  the  first  power  of  .r  if  m  is 
odd,  and  with  the  zeroth  power  of  x  if  m  is  even. 

P,„(.r)  is  the  coefficient  of    the  mth  power  of  z  in  the   development  of 
(1  —  2xz  +  z^)~h   into  a  power  series.    Hence  if   s  <  1 

(1  -  2.r.t  +  .-^)-i  =  Po{x)  +  P,{x).z  +  P,(x).z'  +  P3(.r)..t3 

+  P,{x).z^  +  P,(.r).,.^  +  ■  •  •  +  P,.(.r)..^'"  +  ■  •  • .     (2) 
Whence 

-^====i===^  =  7  r  A(cos  6)  +  ^;  P,(cos  ^)  +  5  P.3(cos  ^)  +  ■  •  • 
V'"  —  2r}\  cos  ^  +  '  1  •-  '  ' 

+  '^'p,„(cos^>+-  --I    if    /■> 

1  r  r  r^ 

=  -     Po(cos  0)+-  Pi  (cos  6)  +  —  P,(cos  0)  +  • 

+  ^' P„,(cos  ^)  +  •  •  •]   if   r<, 

^  =  P„,(.r) 
is  a  solution  of  Legendre's  Equation 

'^ -■'>£- --"I +  "'('" +  !)*  =  « 
when  in  is  a  positive  integer. 

r=  y'»P„^(cOS  e) 

and  F=^^;^^P,„(cos^) 

are  solutions  of  the  form  of  Laplace's  Equation  in  Spherical  Coordinates 
which  is  independent  of  <f>,  namely 

rD;'(rV)  +  ^^I)e(»iriODgV)  =  0.  (4) 

P>na)  =  l-  (5) 

A,«(--'')=^2«(^-)-  (6) 

P.n,^(0)=0.  (8) 

P.(0)^(-r)-^-^;f-,;::^:-^>.  (9) 


Chap.  V.]  TABLE    OF    SURFACE    ZONAL    HARMONICS. 

\-dP....(^^l        ^  .     .yn 3.5.7.  ••  •(2m +  1) 
L        d-r        J,  =  o  2.4.6.  •••2m 


151 

(10) 


For  convenience  of  reference  we  write  out  a  few  Zonal   Harmonics.     They 
are  obtained  by  substituting  successive  integers  for  vi  in  formula  (1). 

T\.(.r}  =  1 

P,(.T)=i(3x^-]; 


^(•^■)-=o(35.r^-30,r^  +  3) 


F^(x)  =  -  (63.rs  -  70.^3  +  15a;) 

P,(,r)  =  ^  (231.T«  -  315.T''  +  105.r2  -  5)  • 

F,(.r)  =  ^  (429.1-"  —  693x^  +  SlS.r^  —  35.x) 

Pg(a-)  :=  ^  (6435a;«  -  12012,r6  +  6930,r'*  -  1260,r-^  +  35). 

Any  Surface  Zonal  Harmonic  may  be  obtained  from  the  two  of  next  lower 
orders  by  the  aid  of  the  formula 

(n  -f  1)P„^  1  (x)  -  (2u  +  l)xP,  (x)  +  nP„_,(x)  =  0  (12) 

which  is  easily  obtained  and  is  convenient  when  the  numerical  value  of  x  is 
given. 

Differentiate  (2)  with  respect  to  z  and  we  get 


(11; 


whence 


(l-2xz  +  z')^ 


{^-'■^) 


P,(x)  +  2P,(^).z  +  3P,(x).z^  + 


Hence  by  (2) 

(1  -  2xz  +  z')(P,(x)  +  2A(a-).^  +  3Ps(x).z'+  ■  ■  •) 

-\-(z-  x){P,(x)  +  Pi(.xO--  +  P-2{x)-^-'  +  •••)=()     (13) 


152  ZONAL   HARMONICS.  [Art.  78. 

(13)  is  identically  true,  hence  the  coefficient  of  each  power  of  z  must  vanish. 
Picking  out  the  coefficient  of  s"  and  writing  it  equal  to  zero  we  have  formula 
(12)  above.* 

78.     We  are  now  able  to  solve  completely  the  problem  considered  in  Art.  9. 
We  were  to  find  a  solution  of  the  differential  equation 

rD,^(r  V)  +  -^^  De(sm  0DeV)=O  (1) 


subject  to  the  condition 
We  know  (v.  Art.  77)  that 


31 

F=— — — -1     when     ^  =  0.  (2) 

(c^  +  H)^  ^  ^ 


V=r"'F„,(cos6) 


and  F=-^,P,„(cos^) 


are  solutions  of  (1). 

Por  values  of   r  <  c 

M              Mr         lv2       1.3  r* 

(6-2  +  ;-^)i  ~  e\_          2  c2  +  2.4  c" 

1.3.5  /•' 

2.4.6  c' 

Therefore  for  values  of   r  <  c 

•]■ 


(3) 


^=f[^o(cos^)-^^>,(cos^) 

+  11  Jl^4(cos  ^)  -|||r_;p,(cos  ^)  +  •  •  •]     (4) 

is  our  required  solution;  because  each  term  satisfies  equation  (1),  and  there- 
fore the  whole  value  satisfies  (1),  and  when    ^  =  0 

P,,^(cos^)  =  P,„(l)  =  l 

[v.  (5)  Art.  77],  and  hence  (4)  reduces  to  (3)  and  (2)  is  satisfied. 
For  values  of    r  >  c 

(c2  _|_  ^2^)1       ,.  |_1       V  ,.2  +  2.4  T^      2.4.6  r'^         J  ^^^ 

\_r      2  r^  ^  2.4  7'^      2.4.6  i-'  ^       J 
*  For  tables  of  Surface  Zonal  Harmonics  v.  Appendix  Tables  I  and  II. 


Chap.  V.]  PROBLEMS    IN    POTENTIAL.  153 

Therefore  for  values  of   r  >  c 
F=  -  I  -  Po (cos  6)--  -3  Po  (cos  6) 

+  H  75  ^*(cos  ^)  -  m  ^.  P.  (cos  ^)  +  •  •  •]         (6) 
is  our  required  solutiou.     For  it  satisfies  (1)  and  reduces  to  (2)  when    ^  =  0 . 

79.  As  another  example  let  us  suppose  a  conductor  in  the  form  of  a  thin 
circular  disc  charged  with  electricity,  and  let  it  be  required  to  find  the  value 
of  the  potential  function  at  any  point  in  space. 

If  the  magnitude  of  the  charge  is  M  and  the  radius  of  the  plate  is  a  the 
surface  density  at  a  point  of  the  plate  at  a  distance  r  from  the  centre  is 

M 


and  all  points  of  the  conductor  are  at  the  potential    -^r—  •    (v.  Peirce's  New- 
tonian Potential  Function,  §  61.) 

The  value  of  the  potential  function  at  a  point  in  the  axis  of  the  plate  at  the 
distance  x  from  the  plate  is  easily  seen  to  be 


Mr 


rclr 


M         ,  a--^  —  «2 


2a  x'  +  a' 

d_/M       _,.r^  — (?A_  3f 

dx  \2a  ^°^      x"  +  aV  ~       a'  +  x" 

_  _  Jf  r    _  a-"       a^*  _  .T®  ~j 

if   a*  <  a, 

^_£ri_^+^_^+ ...1 

a:^L         a-^^x-*       x'^       J 
if    x>a. 
Integrating  and  then  determining  the  arbitrary  constant  we  have 


M 

TT-  COS 


a- MFtt      X 


X        x^  _  *      I    •^■'  ~1 


x-^  +  a^       a  L2 

if   x<ia , 

31 

\_x      3x^  "^  5x^      Ix 
if   a-  >  a . 


164  ZONAL    HARMONICS.  [Ai 

We  have,  then,  to  solve  the  equation 


r  A'('"  n  +  gi^  A  (sin  0  Bo  V)  =  0 


subject  to  the  conditions 

when    ^  =  0    and    r  <  a 

Avhen    ^  :=  0    and    r^  a  . 
The  required  solution  is  easily  seen  to  be 

,,3  1      ,.5 


r 

if    r  <  a    and    ^  < 


-L^--P,(cos^)  +  3-3P3(cos^)-.-,P,(cos^)  +  -J 

nd    ^<J, 

and      r  =  -  f  -  ?,  -.  P"  (cos  6)  +  J  -'  P4  (cos  ^)  -  i  -1  Pg  (cos  ^)  +  ■  •  -I 
if    v>a. 

EXAMPLES. 
1.    Given  that   if  a  charge  M  of  electricity  is  placed  on  an  ellipsoidal  con- 
ductor the  sxirface  density  at  any  point  P  of  the  conductor  is  equal  to     -; — ^  ? 

4:7rabc 

where  p  is  the  distance  from  the  centre  of  the  conductor  to  the  tangent  plane  at 
P  (v.  Peirce,  New.  Pot.  Punc.  §  61) ;  find  the  value  of  the  potential  function  at 
any  external  point  when  the  conductor  is  the  oblate  spheroid  generated  by  the 

rotation  of  the  ellipse  — „  +  7^  ^  1  about  its  minor  axis. 
^      a^      b^ 

Ans.     (1)    If  the  point  is  on  the  axis  of  revolution 
If 


r= 


2\Jir  ■ 


[sin-  (J^L±£^£\  _  sin-  (J!^1^£A£V\ 


X  being  the  distance  from  the  centre. 

(2)     If  the  point  is  on  the  surface  of  the  spheroid 


r=     '' 


2\la^  —  V" 


[f— (^^]=,-^.[f— (v-pb)] 


Chap.  V.]  .    EXAMPLES.  155 


(3)     If  the  distance  r  of  the  point  from  the.  centre  is  less  than   v'^- —  h-  and 
V=  --i=  r?  -  X  -2  ''  z-Ti  A  (cos  6) 


(4)     If  the  distance  r  of  the  point  from  the  centre  is  greater  tlian  \J<r  —  b'^ 


"   V, 


a^  —  b^L        r  3;-^ 


+ 


(^''-H  r>  ....  .^        (1^!-^)^ 


or' 


P4  (cos  ^)  -  ^^^^^^-  Pe  (COS  ^;  +  •  •  -1 


2.    If  the  conductor  is  the  prolate  spheroid  generated  by  the  rotation  of  the 

x^       ir 
lipse     — ,  +  p,  =  1     abont  its  major  axis,  show  that  if  t 

point  and  is  on  the  axis  at  a  distance  x  from  the  centre, 


X         11 
ellipse     — ,  +  p,  =  1     abont  its  major  axis,  show  that  if  the  point  is  an  external 


M       .      x  +  \Ja^-b^ 

V= — ,         -  log , 

2n/«-  —  fr        X  —  >Ja'  —  b^ 


If  the  point  is  not  on  the  axis  and    r  >  \a-  —  b 


..^|I-^=  +  fc^^P.(cos, 


80.  As  a  third  example  we  will  find  the  value  of  the  potential  function  due 
to  a  thin  homogeneous  circular  disc,  of  density  p,  thickness  k,  and  radius  a. 

The  value  of  F  at  a  point  in .  the  axis  of  the  disc  at  a  distance  x  from  its 
centre  is  readily  found  and  proves  to  be 

To  =  2irpk(sjx^  +  a'  —  x)  =  '^  \sjx^  +  a"  —  x']. 
If    x>a 

^•^  +^'^T+:^v  ^•a^  +  2:^^~2:4;?*+2X6.T«  ~2iA8^«+''' 

T  ,,       231  ri  a      1.1^^3  ,   1.1.3  a^      1.1.3.5  r,^   ,        "1 


156  ZONAL   HARMONICS.  [Art.  80. 

It   X  <C  a 

,  ^^  _  2Jf  r .        X    ,1  X^       1.1  .T*        1.1.3  .r6        1.1.3.5  .r'  -| 

and         Fo-—  L        -  +  2^^~2i;^^  +  2.4.6««      2AlIa'^"'_\- 

Hence  the  solution  for  any  external  j)oint  is 

^^      231  ri  a      1.1 «%,       ., 
^=  —     r>  -  —  771  -3  ^2  (cos  ^) 


.  1.1.3  a^  „  ,        ,,       1.1.3.5  r."  „  ,        .,    , 


if   r  >  a,    and 
V 


=  —  ri--Pi(cos6) 

1  r^  1 1  r*  1.1.3  r^  ~| 

+  ^-,A(cos6)-  — -,P,(cos^)+2;^-,Pe(cos^)---J 


73- 
if   r  <  ffi    and    ^  <  ^  ■ 


EXAMPLES. 


1.    The  potential  function  due  to  a  homogeneous  hemisphere  whose  axis  is 
taken  as  the  polar  axis,  is 

if   r  >  a,    and  is 

+  |if>.(cos«)-|iil>.(cos  «)  +  ...] 


TT 

if   7- < «   and    6>  -  ■ 

2.    The  potential   function  due  to  a  solid  sphere  whose  density  is  propor- 
tional to  the  distance  from  a  diametral  plane  is,  at  an  external  point, 

^^       8  3fr5.3a   .   5.3.1  a^  „  ,        ., 

5.3.1.1  a'       ^       ,,    ,    5.3.1.1.3  a'  ^  ^       ^^  -] 


Chap.  V.]  ZONAL   HARMONICS.  157 

3.    The  potential  function  clue  to  the  homogeneous  oblate  spheroid  generated 
by  the  rotation  of    — 2  +  ttj  ^^  ^    about  its  minor  axis  is,  at  an  external  point, 
3        M       r,:^^a^-l^  /  0^^-^^  +  M 

„Va.-2  _|_  ,,2  _  1,2)  J 

if  the  point  is  on  the  axis  of  the  spheroid  at  a  distance  x  from  its  centre. 
y  =  -r^-, 7^1      3-5  ^^ —  TT^  ~ 1 — -  P2  (cos  6) 


if   r>  {fr--l>')^,    and 

if    r<(n-'-h'')^    and    ^<J- 

4.  The  j)otential  function  due  to  the  homogeneous  prolate  spheroid 
generated  by  the  rotation  of  —  +  V:,  ^  1  about  its  major  axis  is,  at  an 
external  point, 

3Jf        r  1    (a'  -  b')l   ,     1    (a'  -  h^)l  „ 

1    (a'-lri 
if    r>  (a^-h^)i. 


+  ^^^^-^'F,(eos&)  +  --^ 


81.  The  method  employed  in  the  last  three  articles  may  be  stated  in 
general  as  follows:  —  Whenever  in  a  problem  involving  the  solving  of  the 
special  form  of  Laplace's  Equation 


rD,^  (/•  V)  +  -7^  Be  (sin  ^  Z»,  F)  =  0  , 
^  sm  $  ^ 


the  value  of  V  is  given  or  can  be  found  for  all  points  on  the  axis  of  X  and 
this  value  can  be  expressed  as  a  sum  or  a  series  involving  only  whole  powers 
positive  or  negative  of  the  radius  vector  of  the  point,  the  solution  for  a  point 


158  ZONAL    HARMONICS.  [Akt.  82. 

not  on  the  axis  can  be  obtained  by  multiplying  each  term  by  the  appropriate 
Zonal  Harmonic,  subject  only  to  the  condition  that  the  result  if  a  series  must 
be  convergent. 

It  Avill  be  shown  in  the  next  article  that  P„j  (cos  6)  is  never  greater  than 
one  nor  less  than  minus  one.  Hence  the  series  in  question  will  be  convergent 
for  all  values  of  r  for  which  the  original  series  was  absolutehj  convergent. 

82.  In  addition  to  the  form  given  in  (1)  Art.  77  for  P,„  (.r)  other  forms 
are  often  useful. 

It  ought  to  be  possible  to  develop  P,„(cos  6),  which  may  be  regarded  as  a 
function  of  0,  into  a  Fourier's  Series,  and  such  a  development  may  be  obtained, 
though  with  much  labor,  by  the  methods  of  Chapter  II. 

The  development  in  terms  of  cosines  of  multiples  of  0  may  be  obtained 
much  more  easily  by  the  following  device. 

We  have  seen  in  Art.  75  that  P^  (cos  6)  is  the  coefficient  of  the  mth  power 
of  z  in  the  development  of  (1  —  2z  cos  6  -\-  z^)~i;  in  a  power  series,  and  that 
if  mod  «  <  1  (1  —  2z  cos  6  +  s^)"^  can  be  developed  into  such  a  series.  We 
know  by  the  Theory  of  Functions  that  only  one  such  series  exists,  so  that  the 
method  by  which  we  may  choose  to  obtain  the  development  will  not  affect  the 
result. 

(1  —  2z  cos  e  +  «2)-i  =  (1  —  z{e.^'  +  «-»')  +  .t-)-7 

^(l-ze^Y^(l-ze-^Y^. 

(1  —  5;e^')-i  may  be  developed  into  an  absolutely  convergent  series  if 
mod  ,t  <  1 ,    by  the  Binomial  Theorem.     We  have 


2.4  '  2.4.6  '  2.4.6.8 


1        „.  ,   1.3   „     „„.  ,  1.3.5   ,     „..  ,  1.3.5 


l-ze-^>)-\=l^^ze-^-^  +  ^z'e-^~^-^^^^z 


zei 


+ 


2  4.6. 


z^e 


The  product  of  these  series  will  give  a  development  for  (1  —  2z  cos  0  +  z^)-  i 
in  power  series.  The  coefficient  of  z^  is  easily  picked  out,  and  must  be  equal 
to    P^(cos^).     We  thus  get 

^„.{cosU)  2.4.6 2m     L       ^         ^2    2^-1*^  ^  ^ 

^2.4    (2//i-l)(2/M-3)<^  ^^  ^^      J 


Chap.  V.]     ZONAL    SUEFACE   HARMONIC    AS    A    SUM    OF    COSINES.  159 

P"  (-^ ") = -Iji^S^^  b '"' '"" + -  ui^,  -^C"  -  2)0 

,  ^         1.3  'm(m  —  1) 
+  ^  1.2(2.M-l)(2m-3J  ^"^("^  -  ^'^ 

,   .,1.3.5  m(m  — l)(m  — 2)  ,        "1 

+  ^  1X3  (2..-l)(2.H---3)(2m-5)  '°^^'"  -  *^^^^  +  ■  •  •  J  •  ^^^ 

If  TO  is  odd  the  development  runs  down  to  cos  6;  if  m  is  even  to  cos  (0),  but 
in  that  case  the  coefficient  of  cos  (0),  that  is,  the  constant  terra,  will  not  contain 
the   factor  2  which   is   common  to  all  the   other  terms,  but  will  be   simply 
ri.3.5---(m-l)~1^ 
L      2.4.6.  •••?«      J' 

We  write  out  the  values  of   P,„  (cos  0)    for  a  few  values  of  m 

Pq  (cos  &)  =  ! 
Pj  (cos  6)  =  cos  e 

p,(cos^)=i(3cos2^  +  l) 

Ps  (cos  e)  =  -(5  cos  3(9  +  3  cos  0) 

P,  (cos  0)=^  (35  cos  40  +  20  cos  20  +  9) 

1  K2) 

Ps  (cos  ^)  =  TTTg  [63  cos  50  +  35  cos  3^  +  30  cos  ^] 

P,  (cos  0)  =  —  [231  cos  m  +  126  cos  4^  +  105  cos  20  +  50] 

P,  (cos  ^)  =  J— ^  [429  cos  7^  +  231  cos  50  +  189  cos  3^  +  175  cos  0] 

P,  (cos  0)  =  ^^T^g^  [6435  cos  80  +  3432  cos  60  +  2772  cos  40 

+  2520  cos  20  +  1225]  . 

Since  all  the  coefficients  in  the  second  member  of  (1)  are  positive,  and  since 
each  cosine  has  unity  for  its  maximum  value  it  is  clear  that  P,„  (cos  0)  has 
its  maximum  value  when  ^  =  0;  but  we  have  shown  in  Art.  76  that  P„,  (1)  =  1. 
Therefore  P„j  (cos  0)  is  never  greater  than  unity  if  0  is  real.  It  is  also  easily 
seen  from  (1)  that    P„j(cos  0)    can  never  be  less  than  —  1. 


160  ZONAL   HARMONICS.  [Art.  83. 

83.     P,„  (.-r)    can  be  very  simply  expressed  as  a  derivative.     We  have 

rJtu  —  1) rJin.  —  ,3)  •  •  •  1  r  m(m  —  1) 

"'^^~  ml  L  2.(2;/i-l) 

m(m-l)(m-2)(m-3)  H 

^     2.4.(2?H  — l)(2m  — 3)  J 

J  ^,„  (X jc^:r  -  ^^^^  _^  ^^  ,  ^x  2.  (2».  -  1)  ^ 

^'    2.4.(2w  —  l)  (2m  — 3)  J 

a'  a;  X 

f'P,,,  (»-)rfa-2  =  Cdx  Cp„^  (x)dx 

^(2m-l)(2m-3)---ir    ,„^.,       (»,  +  2)(m4-l)    , 
(;m  +  2)!  L'  -'-  ^^       ^ 


(m  +  2)(m  +  l)m(».-l)^.,_,  H 


2.(2/«  — 1) 

I  -\-  T)m(m  - 
2A.(2m  —  l(2m  —  3') 

2m(2in  —  1 
(2w)!  L"   '"        2(2m  —  1) 

3m  — 1)  (2m  — 2)  (2. 
2.4.(2m  — l)(2m  — .3) 

-3)---l 

(2m) 


r-p    rr).Z^'" -  (2m-l)(2m-3)---l  F    ,„ _  2.K2m  -  1)  ^.,_, 
J     P„,(.r)r/^    -  L-.  2(2;>.-l) 


(2m-l)(2;.-.3)---l  r    ,„  _  ^^^^,„_,  ^_  n^Q.-l)  ^,_^ 


m(m-l)(m-2)_^.,„_,,   ,         "I 

The  quantity  in  brackets  obviously  differs  from  (x^  —  l)"'  by  terms  involving 
lower  powers  of  x  than  the  mth. 

Hence  P  (x)  -  1-3-5  •  •  •  (2;.  -  1)  ^        _      „ 

Hence  P,„(.^)  -  ^gm)!  f/x- ^^        "^^    ' 

or  P„,  (.r)  =  tJ-t  :^,  (^'  - 1) "'  •  (1) 

This  important    formula  is  entirely  general  and    holds   not  merely  when 
a-  =  cos  6 ,    but  for  all  values  of  x. 


Chap.  V.]     EQUATIONS    DERIVED    FKOM    LEGENDRE's    EQUATION.  161 

84.     The  last  result  is  so  important  that  it  is  worth  while  to  confirm  it  by 
obtaining  it  directly  from  Legendre's  Equation 

(^  -  ^'^  £  "  ^^  I + '"<"' + ^^' = ^  ^^^ 

V.  (1)  Art.  75. 

Let    us    differentiate    (1)    with    respect    to    a;    a   few    times    representing 

^^hyz',^,hjz",-hjz"',&e.     We  get 

(1  -  .T-^)  ^'  -  2.3x  '^'  +  [m(»i  +  1)  -  2  (1  +  2)],t"  =  0 , 

(^  ~  -^"'^  S"  ~  ^-^^  '^'  +  t"^^''^  +  1)  -  2  (1  +  2  +  3)].'"  =  0 , 
and  in  general 

(1  -  .T--^)  ^  -  2(7.  +  l)a^  -^  +  [m(m  +  l)-2(l+2  +  3  +  ---  +  »)]^(">  =  0 

or  (1  -  .r^)  ^'  -  2(n  +  l)x  '^  +  lm{m  +  1)  -  •«(«.  +  1)],.""  =  0  .         (2) 

Following  the  analogy  of  these  steps  it  is  easy  to  write  equations  that  will 
differentiate  into  (1). 

Let    '^  =  z,    ~:  =  z,   '^  =  z,&c.     Then 
dx  dx-  dx^ 


{^-^^')'^.+H^  +  ^>^  =  '^^ 


will  differentiate  into  (1), 

(^  -  "^  &  +  "-^-^  £  +  t^'^^"  +  ^^  -  --^^"^  ^  ^ 
if  differentiated  twice  will  give  (1), 

if  differentiated  three  times  will  give  (1),  and  in  general 

(1  -  x^  '^  +  2(71  -  l)x  ^'  +  [m(m  +  D  -  n(n  -  1)]  z„  ^  0  (3) 

if  differentiated  n  times  with  respect  to  x  will  give  (1). 
If     ?i  =  m  +  1     (3)  reduces  to 

(l-^'^  +  2»..'i^=0,  (4) 


162  ZONAL    HARMONICS.  [Art.  8.5. 

and  the  (m  +  l)st  derivative  with   respect  to  x  of  any  function  of  x  which 
satisfies  (4)  will  be  a  solution  of  (1).     (4)  can  be  written 

(1— '^f-' +  -'"'-«  =  0 

and  can  be  readily  solved  by  separating  the  variables  and  integrating,    v.  Int. 
Cal.  (1)  page  314.     It  gives 

z„^=  C(.r-  — 1)'«. 

Hence  z  =  ^-f  =  C      ^  ,  ,„  (o) 

dx"^  ax'" 

is  a  solution  of  Legendre's  Equation  (1)  and  agrees  with  the  value  of  Pin{x) 
obtained  in  Art.  83. 

85.     The  equations  obtained  in  Art.  84  are  so  curious  and  so  simply  related 
that  it  is  worth  while  to  consider  them  a  little  more  full3^ 
We  have  seen  that 

differentiates  into 

that  if  we  differentiate  (2)  m  times  we  get  Legendre's  Equation 

(1  -  x')  ^,  -  2x  ^,  +  m  (m  +  l).t  =  0 ;  (3) 

that  if  Ave  differentiate  (2)  2m  times  we  get 

(l-^^)£-2(-  + 1)^1  =  0;  (4) 

that  if  we  differentiate  (2)  m  —  n  times  we  have 

(1  -  .r'O  0,+  20^  -  1).>-- 1.  +  i>»{>u  +  1)  -  ni^n  -  l)].t  =  0;  {o) 

and  that  if  we  differentiate  (2)  m.  +  a  times  we  have 

(1  -  x^  '^  -  2(u  +  l)x  f^  +  [m(>n  +  1)  -  n(u  +  1)]-^  =  0.  ((3) 

By  the  aid  of  (1)  we   found   in  the  last  article  a  particular  solution  of  (2), 
namely 


Chap.  V.]      GENERAL   SOLUTIONS   OF   THE  DERIVED   EQUATIONS.  163 

If  we   substitute   in  (2)    z  =  u(x^  —  1)"'   following  the   method  illustrated 
fully  in  Art.  18,  we  get  as  the  general  solution  of  (2) 

,  ^  ^(,..  _  1).  +  B(x^  -  1)"'Jp^,;;T:  '  a) 

A  and  B  being  arbitrary  constants. 

/dx 
— —     ^^^^    is  easily  written  out  [v.  formula  (42)  page  6.     Table  of  Inte- 
(x       1)'" 

grals.     Int.  Cal.  Appendix].     If  x  <  1    it  vanishes  when  a;  =  0.     If  a;  >  1  it 

vanishes  when  .r  z=  oo .     If  then  x  <!  1   (7)  can  be  written 

z  =  A(x'  —  1)">  +  B(x-'  —  1)'"   C 


^  (x^-ir-^ 


and  if   x  >  1 


and  in  these  forms  unnecessary  arbitrary  constants  are  avoided. 
From  (7)  we  can  get  the  general  solutions  of  (3),  (4),  (5),  and  (6). 

(/"(/>•-  — IV"  d'^  r  r        dr         ~1 

dx>"      ^    dxj»L^        ^  J  (x'^  —  iy+^J  ^    ' 

is  the  general  solution  of  (3). 
is  the  general  solution  of  (4). 

^  .^-"(..--1):^ + ^  i!:z!.  r.,. _  ^y.  f__^_i      n 2) 

is  the  general  solution  of  (5). 

is  the  general  solution  of  (6). 

In  each  of  these  forms  A  and  B  are  arbitrary  constants  and  the  integral  is 
to  be  taken  from  0  to  a;  if  ic  <  1  and  from  a;  to  oo  if  a?  >  1. 

Of  course  (10)  must  be  identical  with  the  forms  already  obtained  in  Arts.  16 
and  18  as  general  solutions  of  Legendre's  Equation. 

Equation  (4)  is  so  simple  that  it  can  be  solved  directly,  and  we  get  its 
solution  in  the  form 

^=^''  +  «-/(;^^^.  ("' 

which  must  be  equivalent  to  (11). 


(13) 


164  ZONAL    HARMONICS.  [Art.  85. 

Comparing  (14)  with  (7),  the  solution  of  (2),  we  see  that  every  solution  of  (4) 
can  be  obtained  from  a  solution  of  (2)  by  dividing  the  latter  by  (x^  —  1)'",  or 
in  other  words  that  if  we  write  (2) 

{l~x"Y^^-^2im-l)xj^_-\-2mz  =  0,  (2) 

and  (4)  as  (1  -x^)'^_-  2{m  +  l)^'  ^J  =  0  (4) 

z  =  z-^(x'^  —  1)'"  ;    and  the  substitution  of  this  value  in  (2)  will  give  (4),  and 

the  substitiTtion  of    z-,  =  ^  .,  ~  ^  ^       in  (4)  will  give  ('2). 
(x^  —  iy         ^  ^  ^        ^  ^ 

We  have,  then,  two  ways  of  obtaining  (4)  from  (2) ;  we  may  differentiate  (2) 
2m  times  with  respect  to  x,  or  we  may  replace  z  in  (2)  by  z^ix^  —  1)'". 

If  we  use  the  first  method  we  have  seen  that  Legendre's  Equation  (3)  is 
midway  between  (2)  and  (4).  That  is  if  we  differentiate  (2)  m  times  we  get 
(3)  and  if  we  then  differentiate  (3)  m  times  we  get  (4).  Let  us  see  if  the 
half-way  equation  in  our  second  process  is  Legendre's  Equation. 

If  z  =  y{x^-iyi 

and  y  =  z^^x'^  —  l)f 

z=z,(x-^-ir. 

So  that  if  in  (2)  we  replace  z  by  y(.r"^  —  l)f  and  then  repeat  the  .  operation 
on  the  resulting  equation  we  shall  get  (4).  Making  the  first  substitution  we 
find, 

^1  -  "'>  S  ~  ^^  I + [^^^^'^ + ^>  -  r^J  ^ = ^'         (^^> 

not  Legendre's  Equation  but  a  somewhat  more  general  form.     Of  course  its 
solution  is 

y  =  Aix^  -  l)f  +  Bix-^  -  l)f  J^-^_  .  (16) 

(2)  and  (4)  are  special  forms  of  (5)  and  (6).     Let  us  try  the  experiment  of 

substituting  in  (5)    z  =  y(l  —  x^y\    and  in  (6)    z  =       ^      "  •      We  find  that 

(l  —  xf 

both  substitutions  give  the  same  equation 


Chap.  V.]  ZONAL    HARMONIC    AS    A    PARTIAL   DERIVATIVE.  165 

The  solution  of  (17)  can  be  obtained  from  either  (12)  or  (13)  and  is 

_  1  f  d"'-"{x" 
^-'(l^^iV  dx>-- 

or 


^A^'^^^^^^^^bI^^I^  (IS) 


y  =  a-  r-^f  I  A.  rZ"»-^>'(.r^-l)'«   ,         i;!:^  [     ,  _  ^.  ,„  C ^Ix 1  ^ 

which  of  course  must  be  equivalent. 

86.     In  addition  to  the  value  of  P^i^)  given  in  (1)  Art.  83  there  is  another 
important  derivative  form  which  we  shall  proceed  to  obtain.     It  is 

^..(cos^)  =  ^%-"-'i>/(i).  (1) 

We  have  seen  in  Art.  75  that  -  —  zr  can  be  developed  into 


^l_2^^cos^  +  ^' 


a  convergent  series  if  i\  <i  r  and  that  the  Qn,  -\-  l)st  term  of   that  series  is 

P    (cos  ^)?"i'" 

"^  .,„  +  x •     Let  us  obtain  this  term  by  Taylor's  Theorem. 

11  1  1 


\/l-2-^cos^  +  !l      n/'''-2'VCos^  +  >V^      sjx'^fj^^-2xr,-^r^ 


Regarding  this  as  a  function  of  (x  —  r-^)  and  developing  according  to  powers 
of  i\  by  Taylor's  Theorem  we  get  as  the  (m  +  l)st  term 

Hence  £.l55ii)  =  (^  z)".  (1)  ■ 

87.  We  have  now  obtained  four  different  forms  for  our  zonal  harmonic^ 
a  polynomial  in  x,  an  expression  involving  cosines  of  multiples  of  6,  a  form 
involving  an  ordinary  mth  derivative  with  respect  to  x,  and  a  form  involving 
a  partial  ?/ith  derivative  with  respect  to  x.  We  shall  now  get  a  form  due 
to  Laplace,  involving  a  definite  integral. 

f '!± = !L__^  (1) 

J  ((  —  h  cos  0       (a^  —  b'^y^ 
if   a^  >  h-"    [v.  Int.  Cal.  page  68]. 


166  ZONAL    HARMONICS.  [Art.  87. 

— ; ; — --]  can  be  expressed  in  the  form  —-5 — -1    by  taking  i(=l  —  zx 

(1  —  'J.rs:  +  z%  {a   —  b')^ 

and  l)  =  z  \Jx-  —  1  and  no  matter  what  value  x  may  have  z  can  be  taken  so  small 

that  (/-  Avill  be  greater  than  Ir.     Theii  by  (1) 

1  _\_  r  dcj> _i^  r d^ 

(1  —  2XZ  +  Z^)l  ~  TtJ    1  _  -,,.  _  ;.  \/x-—l.  COS  cj>  "^^    1  —  Z(X  +  sJx-'—l.  COS  c^) 

=:  -    r[l  +  (.r  +  \/x'  —  1.  COS  <^),~  +  (X  +  V.^^^.  COS  </>) V 

+  (•'•  +  y/^-'^^l.  COS  cj^yz^  +  •  •  -yicf, 

if  z  is  taken  so  small  that  the  modulus  of  z(x  +  '^x-  —  1.  cos  <^)  is  less  than  1.    But 
by  Art.  77  (2)  P,„(x)  is  the  coefficient  of  z'"  in  the  development  of , 

^  "•  (1  -  2.7-  +  Z^)\ 

hence  FJx)  =  -  fl^^'  +  V^^' -  1-  cos  <^]'».7c^ .  (2) 

By  replacing  c^  by  tt  —  (f>  in  (2)  we  get 

n.(-^-)  =  ^ /"[■'■  -  V-^^^^l.  e(.s  <^]"v7c^.  (3) 

and  if  mod  -  <  1  or  in  other  words  if 


mod  ..  .^  1  ^  ^^j^  l^g  developed  into  a  convergent  series  involv- 

ing  powers  of  -,  and  the  coefficient  of  (-)     will  be  FjJ-'^);    but  this  will  be 
z  \z/  -^ 

the  coefficient  of  z-"'~^  in  the  development  of    — >,  _  _i_    -771    according  to 

descending  powers  of  z,  mod  z  being  greater  than  1. 

If  now  we  let    a  =zzx  —  1    and   b  =  z  \x^  —  1,    a'^  —  //-  =  1  —  2xz  +  s:'^   and 
z  may  be  taken  so  great  that    a^  —  b^X).     Then  by  (1) 

1  _  1  ^ d4> 

(1  —  2./-.V  +  «2)4-  "~  ttJ  ....-^.  —  1  —  z  ^x-'  —  1.  cos  <^ 


-^/- 


r/c^ 


^(a;  -  V,r-  -  1.  cos  c/))  fl ,    "^  1 

L         z(x  —  >^x^  —  l.coscf>)-i 

=1  f ^i r.-+ =L ,.- 

V  C^'  —  V^.^-'  —  1-  cos  <^)  L  (x  —  ^x^  —  1.  cos  <^) 

1 

(.X  —  ^x'  —  1.  cos  <l>y 


+ ',^^. r.  -'  +  •  •  •]  f^<^ 


Chap.  V.]  DEVELOPMENT    IN    ZONAL    HARMONIC    SERIES,  167 

and  the  coefficient  of  z"'"-'^  is     —  ( 

ttJ  r 


[x  —  V^"^  —  1.  cos  c^]" 
Replace  ^  by  tt  —  <^  and  we  get 


Hence  -  ,„,    y  .  , 

[x  —  \x"  —  1.  cos  </)] 


p™(^-) = -  r — p=^^^ — —  •  (5) 

88.  In  the  problems  in  which  we  have  already  nsed  Zonal  Harmonics 
(v.  Arts.  78-81)  we  have  been  able  to  start  with  the  value  of  the  Potential 
Function  at  any  point  on  the  axis  of  A',  and  it  has  been  necessary  to  develop 
the  expression  for  V  on  that  axis  in  terms  of  ascending  or  descending  powers 
of  X.  If,  however,  we  start  with  the  value  of  V  in  terms  of  6  for  some  given 
value  of  r,  that  is  on  the  surface  of  some  sphere,  we  must  develop  the  function 
of  0  in  terms  of  zonal  harmonics  of  cos  6  (v.  Art.  10),  and  our  problem  becomes 
the  following :  —  To  develop  a  given  function  of  cos  0  in  terms  of  zonal  har- 
monics of  cos  6,  or  to  develop  a  given  function  of  x  in  terms  of  the  functions 
P„,(a-),  X  lying  between  1  and  —  1. 

The  problem  resembles  closely  that  of  developing  in  a  Fourier's  series, 
which  we  have  already  considered  at  such  length. 

Let  /(:.■)  =  A,P,(x)  +  A,P,(x)  +  J,P,(.r)  +  A,P,Qc)  +  ■••  (1) 

for  all  values  of  x  from  —  1  to  1  and  let  it  be  required  to  determine  the 
coefficients. 

If  f(x)  is  single-valued  and  has  only  finite  discontinuities  between  x^  —  1 
and   x^l    we  may  proceed  as  in  Art.  19. 

Let  us  take  n  -\-  1  terms  of  (1)  and  attempt  to  determine  the  coefficients. 
Take  7i-\-l  values  of  x  at  equal  intervals  Ax  between  x  =  —  1  and  x=^l 
so  that  (u  +  2)\x  =  2-  /(-l-fA,T),  f(-l+2Ax),  f(- 1  +  SAx),  -  " 
/[ — l  +  («  +  l)Aa,"]  will  be  the  corresponding  values  of  f(x).  Substitute 
these  values  in  (1)  and  we  have 

/(-  1  +  Ax)  =  AJ\(-  1  +  Ax)  +  A,P,(-  1  +  Ax) 

+  A,P,(-  1  +  Ax)  +  •  •  •  +  J„P„(-  1  +  Ax) 

fi-  1  +  2A.r)  =  JoPo(-  1  +  2  Ax)  +  A,P,{  -1+2  \x) 

+  A,P,(-  1  +  L'Ar)  +  •  •  •  -f  A„PJ-  1  +  2Aa.) 

JXI  -  Ax)  =  A,P,a  -  A,r)  '+  A,P,(1~  Ax)  +\-UP,(l  -  Ax)  +  ••• 

+  ^„P„(l-Ax-),j 

that  is,  n  -{-  1  equations  from  which  in  theory  the  n  -\-  1  coefficients 
Aq,  Ai,  •  •  •  A^  can  be  determined. 


h(2) 


168  ZONAL   HARMONICS.  [Art.  89. 

Following  the  analogy  of  Art.  24  let  us  multiply  the  first  equation  by 
P„;(— 1  +  A.r).Aa;,  the  second  by  P,„(— 1  +  2Aa:;).Aa; ,  the  third  by 
P„,(— 1 +3A.r).Ax,  &e.,  and  add  the  equations.  The  first  member  of  the 
resulting  equation  is 

2^/(-  1  +  kAx)F,„{-  1  +  k^x).Ax  ,  .      (3) 

k=  1 

and  the  coefficient  of  any  A  as  Aj  in  the  second  member  is 
i-  = « + 1 
2)P,„(-  1  +  kAx)P,(-  1  +  k\x).Ax.  (4) 

If  now  n  is  indefinitely  increased  (3)  approaches  as  its  limiting  value 

yf(x)P^(x)dx  (5) 

—  1 

and  (4)  approaches  |  P^(cc)P,(a;)(Za;.  (6) 

We  have  now  tp  find  the  value  of  the  integral  (6)  or  as  we  shall  write 
it  for  the  sake  of  greater  convenience 

jF^(x)P,(x)dx. 

89.  rP.(.)P„(.)^.  =  ,^^J-r-,  r^!(^J):  .  ^V-1)"  dx 

-1  -1 

by  (1)  Art.  83. 

rdr\x^  -  \Y  d^x'—iy  ^j^  ^  rrZ"'(.r^  -  1)"'  d"-\x^-iY-\ 

J         dx"^  dx"  L        dx"'  dx"~^        J 

—  1  —1 

by  integration  by  parts. 
Now  it   z  =  X(x^  — 1)» 

I  =  2nxX(x^  -  1)"-  +  (x^  -  ly  ^=  (.r^ -  1)'-^  [2nxX+  (x^  -  1)  g]  •  (2) 

Hence  the  ^th  derivative  with  respect  to  x  of  any  function  of  x  containing 
(x^  —  1)"    as  a  factor  will  contain   (x^  —  1)" ~^   as  a  factor  if  p<n. 


Chap.  V.]  DEVELOPMENT    IN    ZONAL    HARMONIC    SERIES.  169 

^  ^_^ — —  •     then,  contains    (x'^  —  1)   as  a  factor  and  is  zero  when    x  =  l 

and  when   x  =  —  1  ,    so  that  (1)  reduces  to 

''^'"(■'■'-1)"  'ffr— D- ,,,  _ _  ri-\.,-'-ir  d—(^'-iy  ^^^ 


J         dx"'  dx"  J         dx"'  +  ^ 

—1  —1 

It  follows  that 

J  dx"'  dx"  ^        '  J  d:c"'  +  ''  dx''-^ 

-1  -1 


ri'x^"'  r/.r"-"' 


If    //i  <  w    we  get  from  (.S) 

If      111  >  ?? 

If,  then,  ??i  is  not  equal  to  n 

j'p„,(x)P,(x)dx  =  0.  (4) 

—  1 
1 

If    vi=zn    we  have  to  find      ^[^P,J^(x)Yd^• 
-1 

by  (3),  =  (—  l)"'(2/«) !  C{x^  —  lYdx. 


170  ZONAL   HARMONICS.  [Art.  90 

1  1  ^ 

C(x'  -  i)"'dx  =  C(x  -  i)'"(,r  +  ly^dx  =  -  £^f(x  -  iy'-\x  +  ly^+hix 


-(-i)"'(..  +  ,)e^:+,).r7^(--  +  i)-^ 


=  (-ir 


2"'  +  hn\ 


(m  +  l)(m  +  2)---(2m-^l) 


Hence  ClP„,(;x)Ydx  =  ,^,„,/  ,,.,  /     !\::    liL        !^      i  in 

—  I 

90.     The   solution  of  the  problem  in  Art.  88  is  now  readily  obtained,  and 
we  have 

f(x)  =  APo(*)  +  A,P,(x)  +  A,P,(x)  +  ■■■  (1) 


where  A,^ 


=  ^-^fmP.(^')dx.  (2) 


The  function  and  the  series  are  equal  for  all  values  of  x  from  x^  —  1  to 
x  =  l,  and  f(x)  is  subject  to  no  conditions  save  those  which  would  enable  us 
to  develop  it  in  a  Fourier's  Series,     [v.  Chapter  III.] 

Of  course  (1)  can  be  written 

/(cos  0)  =  A,F,(cos  0)  +  ^iPi(cos  6)  +  ^2Po(cos  6)^ 

where  A,,  = ^  J  /(cos  0)P,„ (cos  6)^Z(cos  6) 

or  if  /(cos  6)  =  F(0) 

F(d)  =  AoPo(cos  6)  +  ^1  Pi  (cos  6)  +  A  ^2  (cos  0) -\ (3) 


rhere 


2  m  +  1 


A^  CF(e)P, „(cos  0)  sin  e.de  (4) 


and  the  development  holds  good  from    ^  =  0    to    0  =  7r. 

If  f(x)  is  an  even  function,  that  is,  if   /(— a')=/('^)    (1)  ^^^  (2)  can  be 
somewhat  simplified.     For  in  that  case  it  can  be  easily  shoAvn  (v.  Art.  77)  that 


pXx)P,,(x)dx  =  2pXx)P,,(x)dx, 


Chap.  V.]  DEVELOPMENT    IN    ZONAL    HARMONIC    SERIES.  171 

an  d  that  |  f(x)  P^^.  +  j  (x)  dx  =  0; 

-1 
so  that  if  /(—  x)  =f(x) 

fix)  =  A,Po(x)  +  A,P,(x)  +  A,P,(x)  +  J,P,(x)  +  ■"  (5) 

where  J,,  =  (4k  +  1  )ff(x)P,,(x)dx .  (6) 

If  f(x)  is  an  odd  function,  that  is,  if  /(—  x)  =  — f(x)  it  can  be  shown  in 
like  manner  that 

f(x)  =  A,P,{x)  +  A,P,i.r)  +  A,P,i.r)  +  A,P,{x)  +  •  •  •  (7) 

where  A,,^,  =  (4k  +  3)jf(x)P,,^,(x)dx.  (8) 

If  it  is  only  necessary  that  the  development  should  hold  for  0  <  a;  <  1  any 
function  may  be  expressed  in  form  (5)  or  (7)  at  pleasure. 


a  more  gen- 


91.     We  can  establish  the  fact  that      Cp„^(x)P,^(x)dx  =  0    by 

eral  method  than  that  used  in  Art.  89. 

Let  A'„j  be  any  sokition  of  Legendre's  Equation 

jx[_^^~  -'"^  £] + '"^'" + ^>' = ^     [^^-  ^^>  ^'^'-  ^^^' 

which  with  its  first   derivative   with  respect  to  x   is   finite,  continuous,  and 
single-valued  for  values  of  x  between  —  1  and  1,  —  1  and  1  being  included. 

'"'™        £[(i-->'-f']+'»("'+i)-^-=«  (1) 

Multiply  (1)  by  A'^  and  (2)  by  X„^  and  subtract  and  integrate  and  we  get 

lm(m  +  1)  -  n(u  +  l)-]fx^XJx  =fx,,  £  [(1  -  ^')  '^'^x 

—  1 


172  ZONAL   HARMONlCSo  [Art.  91. 

Integrate  by  parts, 
lm(m  +  1)  -  »(«  +  l)]j'x„X,cb  =  [j.i;.(l  -='')^-  X.(l  -  »•»)  !^-]" '_ ^ 


(^ic     dx 


Whence  Cx,„X„dx  =  0  (4) 

—  1 
unless    m  =  n. 

(3)  gives  at  once  the  important  formula 

r,,,    a-^-)[-v/i°'-x„f-] 

from  which  come  as  special  cases 

(i-.^[p.(.)^-p„(.)^] 

J  P„(.x)A(.T),fa  = ,„(„,_m_„(„  +  i) (6) 

and  since    Po(ic)  =  1 

unless    m  =  0 . 

EXAMPLES. 

1.     Show  that      I  P^,(x)dx  ^0     if  vi  is  even  and  is  not  zero. 


(-1)    -     ...u..A_^^OAa   ....^_ix       if 


J^ 3.5.  <■  •••m 

m(m  +  1)  2.4.6.  •  •  •  (m  —  1) 
odd.   V.  Art.  91  (7)  and  Art.  77  (10). 
2.     Show  that 


Cp^{x)P„{x)dx  =  Q     if 


m  and  ??.  are  both  even  or  both  odd. 


(-1)' 


,  t ./.  I 


2'"-^"-y(n^  -  n)(vi  +  n  +  l)^!)  (^l) 


if  m  +  ?i  is  odd.     v.  Art.  91  (6)  and  Art.  77  (8),  (9),  and  (10).     cf.  J.  W. 
Strutt  (Lord  Kayleigh)  Lond.  Phil.  Trans.  1870,  page  579. 


3.     Show  that      ClP^^{x)Jdx  = I v.  Art.  89  (5) 

TT  2w  +  1 


Chap.  V.]  DEVELOPMENT    IN    ZONAL    HARMONIC    SERIES.  173 

92.  Formula  (4)  Art.  91  can  be  obtained  directly  from  Laplace's  Equation 
by  the  aid  of  Green'' s  Theorem  (v.  Peirce's  Newt.  Pot.  Punc.  §  48). 

Take  the  special  form  of  Greeiv's  Theorem  [(148)  §  48  Peirce's  Newt.  Pot. 
Func] 

fff(^  V'  V-VV^  U) dxchjdz  =J(  UI)„  V  -  VD„  U)ds  (1 ) 

where  V^  stands  for  {D^  +  D^  +  D^),  D^  is  the  partial  derivative  along  the 
external  normal,  and  the  left-hand  member  is  the  space-integral  through  the 
space  bounded  by  any  closed  surface,  and  the  right-hand  member  is  the  surface 
integral  taken  over  the  same  surface,     (v.  Int.  Cal.  Chapter  XIV.) 

If  ?7and  F  are  solutions  of  Laplace's  Equation  V^F=V'^f^=0  and  (1) 
reduces  to 

^{  UI),^  V  -  VD,  U) ds  =  0 .  (2) 


/ 

Now  r"'X„^  and  r"Z„  are  solutions  of  Laplace's  Equation  if  .r  =  cos  $ 
(V.  Art.  16). 

If  the  unit  sphere  is  taken  as  the  bounding  surface  and  U  =  r"'A',„  and 
F=  r"X^    (1)  and  (2)  will  hold  good. 

Z)„  U  =  D,.(r"XJ  =  w.r'"  -  'X^ , 

D„V=nr"-KY„, 

ds=z  sin  0.dedcf>, 

and  (2)  becomes         Cdcf,  C(nX„^X„  —  mX^^^XJ  sin  e.dO  =  0 

0  0 

or  27r(w  —  m)  rA''„,A;,  sin  d.de  =  0  .  (3) 

0 

Since    x  =  cos  6 ,  sin  6.d0  ^=  —  dx   and  (3)  reduces  to 

'x^X^dx^O*  (4) 


P 


unless    m=:n. 

93.  We  can  now  solve  completely  the  problem  of  Art.  10  which  was  in 
that  article  carried  to  the  point  where  it  was  only  necessary  to  develop  a 
certain  function  of  0  in  the  form 

AoPo(cos  6)  +  .4iPi(cos  6)  +  A.P^icos  e)-\ 

*  It  should  be  noted  that  this  proof  is  no  more  general  than  that  of  the  last  article,  for,  in 
order  that  Green's  Theorem  should  apply  to  r"'X„;,  this  function  and  its  first  derivatives  must 
be  finite  continuous  and  single-valued  within  and  on  the  surface  of  the  luiit  sphere,  (v.  Peirce, 
Newt.  Pot.  Func.  §  48.) 


174  ZONAL   HARMONICS.  [Art.  94- 

given  that  f(6)  =  1    from    ^  =  0    to    6  =  '^ 

and  f(0)  =  0   fi'om    ^  =  ^   to    O^tt. 

This  amonnts  to  the  same  thing  as  developing  F(x)  into  the  series 
F(x)  =  A,P,(x)  +  A,P,(x')  +  A,P,(x)  +  AsP,(x)  +  ■■• 
where  F(x)  =  0    from    x  ^  —  1    to    .r  =  0 

and  F(x)  =  l    from   ■<■  =  0        to    .r  =  l. 

By  Art.  90  (1)  and  (2) 

2m  +  1  / 
and  any  coefficient  -•i/n  =  — 7) — "  I  ^mi^')'-^^' 

By  Art.  91,  Ex.  1 

I  Pj^(x)dx  =:  0      if  7n  is  even 

0 

^     ^'^         1  3.5.7.  •'•m  •£        ■      jj 

^       ^         //i(??i  +  1)  2.4.6.  •  •  •  (ill  —  1) 

Hence  A,,^  =  0     if  7n  is  even 

^         '^2>/^+l    1.3.5.  •  •  •  (m  -  2)     .„       ... 
=  (-1)^    2^+2- 2.4.6.  •••(m-1)     ^f--«°dd. 

Then  i.(..)  =  1  +  ^  P,(.r)  -  1. 1  AC^O  +  ^-  H  Ps(r)  -■■■  (1) 

and       «-=  ^  +  ^  rP,(cos  ^)  -  ^ •  ^  /•^P3(cos  ^)  +  ^  •  ^  r^PsCcos  6)  +  ■  ■  ■        (2) 
for  any  point  Avithin  the  sphere. 

94.  If  in  a  problem  on  the  Potential  Function  the  value  of  V  is  given  at 
every  point  of  a  spherical  surface  and  has  circular  symmetry*  about  a  diameter 
of  that  surface  the  value  of  V  at  any  point  in  space  can  be  obtained. 

We  have  to  solve  Laplace's  Equation  in  the  form 

rD,\i- V)  +  gA-^  A(sin ei)eV)=0  (1) 

*  See  note  on  page  12. 


Chap.  V.]  POTENTIAL   GIVEN   ON   A   SPHERICAL   SURFACE.  175 

subject  to  the  conditions 

V=f{6)    when    rz=a 

F=0  *'        ;-=oo. 

We  have        f{d)  =  AoPo{cos  0)  +  -4iPi(cos  0)  +  A.P.(<io8  6) -\ 

where  A^  =  — ^-~  j /(^)P^(cos  6)  siu  6.d0.  v.  Art.  90  (4). 

Hence 
F=  A,  +  A,  (-^')  Pi(cos  ^)  +  A,  (^")'a(cos  ^)  +  A,  0)'a(cos  ^)  +  •  •  •    (2) 

is  the  required  sohition  for  a  point  within  the  sphere,  and 

V=  A,  0)  +  A,  0)  Pi  (cos  6)  +  A,  (^)  P2(cos  0)  +  A,  (^)  ^3(008  ^)  +  •  •  •  (3) 

is  the  required  solution  for  an  external  point. 

EXAMPLES. 

1.  If  on  the  surface  of  a  sphere  of  radius  c  V  is  constant  and  equal  to  a 
show   that     V=a     for   any   point  within    the   sphere    and    F= —    for   any 

external  point. 

2.  Two  equal  thin  hemispherical  shells  of  radius  c  placed  together  to  form 
a  spherical  surface  are  separated  by  a  thin  non-conducting  layer.  Charges  of 
statical  electricity  are  placed  on  the  two  hemispheres  one  of  which  is  then 
found  to  be  at  potential  a  and  the  other  at  potential  b.  Find  the  value  of  the 
potential  function  at  any  point. 

F=^-+(6-a)[^^P,(cos^)-^.^^P3(cos^) 


,11    1.3  >%^ 

+i2-2:4;r^^^^^°' 


for  an  internal  point 


V= 


■■  ;.  +  (^  -  «')  [I  7.  A(cos  0)-l.l'^^  P3(cos  0) 
for  an  external  point. 


176  ZONAL    HAinrONICS.  [Art.  94. 

3.  If    Fi  =/(cos  6)    when    r  =  a    and    V^  =  0    when    r  =  h    show  that  for 
a  <r<b 

2m  4-  1  / 
where  J,„  =  — j^ — J  f{.r)P„,(x)dx . 

—  1 

4.  If    Fa  =^  F(cos  6)     when     r  =  b     and    I'a  =  0     when     /■  :^  a     then  for 
r/  <  /■  <  Z* 

n=|^,.e:-':^:)(5:-F^:)""^«(eos.)  . 

2m  -I-  1     /i 


where 


5.  If  the  value  of  the  potential  function  is  given  arbitrarily  on  the  surfaces 
of  a  spherical  shell  but  has  circular  symmetry  *  about  a  diameter  J'^  V^  +  V.2 
(v.  Exs.  3  and  4). 

6.  Two  concentric  hollow  spherical  conductors  are  insulated  and  'charged. 
The  inner  one  of  radius  a  is  at  potential  7>,  and  the  outer  one  of  radius  b  is  at 
potential  q.     Find  V  for  any  point  in  space. 

V^2^     if     r<a, 

b  —  (I  \r         /        b  —  a  \         r/ 

V='^     if     r>b. 
r 

7.  If  r^O  on  the  base  of  a  hemisphere  and  7'=/(cos^)  on  the  convex 
surface,  show  that  for  a  point  within  the  hemisphere 

where  A,,^,  =  (U  +  S)Jf(x)P,,^,(x^dx  [v.  Art.  90  (8)]. 

8.  If  the  convex  surface  of  a  solid  hemisphere  of  radius  a  is  kept  at  the 
constant  temperature  unity  and  the  base  at  the  constant  temperature  zero 
show  that  after  the  permanent  state  of  temperatures  is  set  up  the  temperature 
of  any  internal  point  is 

n  =  l^  P,(cos  &)-l-l$  P3(cos  ^)  +  f  •  II  ^;  A(cos  0)  -  ■  ■  ■ 
*  See  note  on  page  12. 


Chap.  V.]  DEVELOPMENT    OF    X".  177 

9.  A  sphere  of  radius  a  and  with  blackened  surface  is  exposed  to  the  direct 
rays  of  the  sun  in  air  at  the  temperature  zero.  Find  the  stationary  tem2)erature 
of  any  internal  point. 

Suggestion:    D/ii  -\-  hx  —  Mf(6)  =  0    when    r  =  a . 

Let  M  =^^,„  ^  P„,(cos  ^)  ,     and    f(0) -=^B„^Pjcos  $). 

Then  we  have 

^7^. ^  P„(cos  6)  +  h ^^,„P,„(cos  6)  -  M ^B,,P^(cos  e)=0, 

whence  ^,„  =  — — ~  ■ 

h  +  - 

Here    fiB)  =  cos6   if   0  <  ^  <  ^   and   /(O)  =0    if   ^  <  ^  <  ^^ . 

/(^) =1+1  ^i(««s  ^)+T6  ^^^''''  ^^  ~  i  ^*^^°'  ^^  +  ■  ■  ■ 

+  (-  i).-.i (^Jc  +  l)(2k)l 

^  ^     ^^        (47,  +  4) (2k  -  l)2-'\k \y  ^^^<'''''  ^^  + 

V.  Art.  91  Exs.   (2)  and  (3).     cf.  J.  W.  Strutt  (Lord  Rayleigh),  Loud.  Phil. 
Trans,  vol.  160,  page  587. 

95.  The  formulas  of  Art.  90  enable  us  to  develop  a  given  function  of  x  in 
terms  of  Zonal  Surface  Harmonics,  the  development  holding  true  for  values  of 
X  between  —  1  and  +1.  If,  however,  we  can  show  by  outside  considerations 
that  a  given  function  of  x  can  be  expressed  in  Zonal  Surface  Harmonics,  the 
development  holding  true  for  all  values  of  x,  the  formulas  of  Art.  90  will  give 
us  the  development  in  question. 

For  example  if  n  is  a  positive  integer  x"  can  be  expressed  in  terms  of  Zonal 
Surface  Harmonics  no  matter  what  the  value  of  x,  and  no  Harmonic  of  higher 
order  than  n  will  enter.  For  the  formulas  giving  the  values  of  Pi(x),  P2{x),  •-• 
P„(x)  (v.  Art.  77)  may  be  regarded  as  n  algebraic  equations  of  the  first  degree 
in  terms  of   x,  x^,  x^,  •••  ic"   and   Pi(x),  P2,{x), •  •  •P„(a;). 

From  these  equations  the  n  —  1  quantities  a?,  cc^,  a;^,  •• -a;""" ^,  can  be  elimi- 
nated, and  there  will  result  an  equation  of  the  first  degree  in  x^  and  Pi(x), 
P2(x),  •  •  •  Pn{x),  which  will  enable  us  to  express  x"  in  the  form 

A,  +  .liPi(a-)  +  A,P,{x)  +  •  •  •  +  A,P,(x) , 

no  matter  what  the  value  of  x,  and  we  shall  have  the  same   formula  when 
—  1  <  .T  <  1    as  when    .x  >  1    or   a;  <  —  1. 


178  ZONAL   HARMONICS.  [Art.  9.5. 

Let  us  obtain  this  development.     By  Art.  9(»  (1)  and  (2) 

X"  =  A,P,(x)  +  A,P,(x)  +  A,P,(x)  +  •  •  •  (1) 

where  A^  =  — ^  \  x" P „^{x)dx .  (2) 

—  1 

A„  =  '-^  ^J.>  '^^iBLr^  ^  by  (1)  A..  83. 

—  1 

By  integration  by  parts  we  get 

r^"  ^  "7  .7       dx  =  n(n  —  l)(w  —  2)  •  •  •  (n  —  »i  +  1)  fa;"- «(1  —  x^'dx ,   (3) 
—1  —1 

if    VI  <C  n  -\-  1, 

=  0    if    m>u. 
By  integration  by  x^arts  we  readily  obtain  the  reduction  formula 
1  ^         1 

ra;^(l  -  xydx  =  -^^  C^"  ^  '(1  -  ^^y  ~ ^dx  whence 

Ln-mn  - ^yndx  = i^wi! r    . ,„  , 

r  2  .  . 

I  »■"  +  "'cZa;  =  — ; r—.      if     n  +  ?/i     is  even  , 

J  n-\-m-\-l 

==  0     if     n  +  wi     is  odd. 


Hence         A, 


Therefore 


(2m  +  l)n{n  —  1) fa  —  2)  •  •  •  (w  —  m  +  1) 
(m  — m  -\- 1)  (?i  —  7)1  -\-  3)  (?i  — m  -\-  5)  •  •  •  (11 -\-  m  +  1 

if    »i  <C  /?  4*  1    and    ?/?-  +  n   is  even, 

=  0     if    m  >  /«    or  if    m  -\-  n    is  odd. 


""  =  1.3.0  ■■■p.t  +  l)  [(2"  +  l>^"('">  +  (2»  -  3)  ^^^^  ^.-,(-») 
+  (2,.-7)<^"+^f-^^P._.(.) 


ending  witn  tne  term     — 
n- 


the  second  member  ending  with  the  term     — r— ,  Pq(x)     if  n  is  even  and  with 
o  ?i  + 1       ^  ^ 


3 
the  term     — r— r  Pxix)     if  /i  is  odd 


Chap.  V.]  USEFUL   FORMULAS.  179 

For  convenience  of  reference  we  write  out  a  few  powers  of  x. 

X  =  P^(x) 

1 


,2^ 


-A(.t)  +  -Po(x) 


^'  =  q:^Ps(x)+^Ps(x)  +  ^P,(x) 

16  24  10  1 

^'  =  231  ^«^^'^  +  77  ^^("""^  +  21  ^'^''^  +  ^  ^°^^^ 


(5) 


^^  =  Mk  ^«(-^)  +  H  ^«(^->  +  ll  ^^(^-^  +  9-9  ^^('^^  +  -9  ^  o^-J 

If  a  given  function  of  x  can  be  expressed  as  a  terminating  power  series  it  can 
be  developed  into  a  Zonal  Harmonic  Series  by  the  aid  of  (4).     Given  that 

/(,r)  =ao  +  ((iX  +  a.x-  +  (laX^ -\ . 

let  f(x)  =  B,  +  7AP,(.r)  +  P,P,(x)  +  B,P,(x)  +  •  •  • ; 

then  picking  out  carefully  the  coefficient  of  P,„(^)  we  have 

(m  4-  1^ 

«.  +  .+  •••     •   (6) 


'"       1.3.5.  •  •  •  (2m  —  1) 

(w  +  l)(m  +  2)(m  +  3)(m  +  4) 


2.4.(2m  +  3)(2m  +  5) 
96.     The  development  of   — y-^    is  useful  and  is  easily  obtained. 
Let  '^l^  =  A,P,{x)  +  A,P,{x)  +  A,P,{x)  +  •  •  • 

Then  ^,„  =  ^1JP,„^,.)^,.  (1) 

by  Art.  90  (2) ; 


180  ZONAL    HARMONICS.  [Art.  97. 

[P,„(.r)P„(*)f  ~      =  0    if    VI  +  n    is  even 

=  2    if    m  +  n    is  odd. 

(IP  (x) 
Since  P„(x)  is  an  algebraic  polynomial  of  the  nth.  degree  in  x,    — y^-^  is  an 

algebraic  polynomial  of  tlie  n  —  1st  degree  in  .r.     Therefore  in  (1)  m  is  less 

dP  (x) 
than  n;  consequently   — p"-^   is  an  algebraic  polynomial  iu  x  of  lower  degree 

than  n  and  i 

Cp„{x)  '^^f^  dx  =  0  by  Art.  95  (3). 

—  1 

"We  get  then  J,„  =  2 in  +  1    if    ni  +  n    is  odd  and    m  <  n, 

==:  0    if    ni  -\-  n    is  even  or    m  '>  n  —  1  ;  and 

^^^  =  (2n  -  l)P„_i  {x)  +  (2;.  -  o)P„_,{x)  +  (2m  -  9)P„_,(^)  +  •  •  •  (3) 

the   second  member  ending  with  tlie   term   ^Pi{x)  if  n  is  even  and  with  the 
term  Pq{x)  if  n  is  odd. 

From  (3)  a  number  of  simple  formulas  are  readily  obtained.     For  example 

jP.{:c)dx  =  2^  IP,  _,(.,■)  -  P„  ,  ,(»•)]  .  (6) 

X 


[v.  (4)  and  Article  77  (12)]. 


r7P  /^^ 


[v.  (5)  and  Article  91  (7). 

97.     By  the  aid  of  the  formulas  of  Art.  96  a  number  of  valuable  develop- 
ments can  be  obtained. 

Let  us  get  cos  nO  and  sin  nO  n  being  any  positive  real. 

z  =  cos  7i6   and   z  =  sin  nO   are  solutions  of  the  equation  , 

^^^  _L       2  A 

— +  n2^  =  0 


Chap.  V.]  ADDITIONAL    DEVELOPMENTS.  181 

or  if  we  let   x  =  cos  6,    of  the  equation 

Let  aoA(*0  +  <hPi(x)  +  «2A0>')  H 

be  the  required  development  of  cos  n6  or  of  sin  7i0. 


Then 


X  «»  [(1  -  -')  ^^^  -  -  ^^  +  -^P.(-)]  =  0  by  (1). 


=  F„^(x)  is  a  solution  of  Legendre's  Equation  (v.  Art.  77).     Hence 
,,  (PF,Joc)  dP„,(x)  dF,Jx)  ,   , ,  ^       , 


dx^  dx  dx 

and  (1)  becomes 


(2) 


Formulas  (4)  and  (6)  of  Art.  96  enable  us  to  throw  (2)  into  the  form 

^"'\_2?n-i-l         dx  2/«  +  l  dx        J~^-  ^'^^ 

dP       (x) 
(3)  must  be  identically  true.     Therefore  the  coefficient  of    — "'T  must 


equal  zero,  and  we  have 


^"l'  I  ^  I"  III'  ,    ,    N 


2m  -{-  5         91^  —  )/i 
If  we  are  developing  cos  n6 

ao  =  -  jcos  nO  sin  e.cW  by  Art.  90  (4), 

0 

=  -  ffsin  (w  +  1)^  —  sin  (n  —  l)$2de  , 

0 

11+  cos  nTT 


and 


2        n'-  —  1 
3 


(5) 


«!  =  -  j  COS  ?«.^  cos  (9  sin  d.dO  by  Art.  90  (4), 

(6) 


3    1  —  COS  mr 


182  ZONAL   HARMONICS.  [Art.  97. 

(4).  (5),  and  (6)  give  us 

COS  nO  =  -  —^^7^ T-     Po(cos  ^)  +  5  -. ;-,  PJcos  6) 

2[n-  —  1)    L  n^  —  o-       ^ 


^^  [3P,(eos  6)  +  7  '^,  P.(cos  6) 


1  —  COS  mr 

2(n^  —  2^) 


r   or    1  —  COS  ', 
^„(cos^).     For 


If  %  is  a  whole  number    1  +  cos  7i7r   or    1  —  cos  mr    will  vanish  and  the  series 
will  end  with  the  term  involving  P„(cos  6).     For  this  case  (7)  maybe  rewritten 


+  (2»-3)5^r{~ly>.-.(«»^«) 


^  ^  [)r — (n — 2)^]['/r — (>i  ~  Vj  J 

If  we  are  developing    sin  nO 

ao  =  -  I  sin  n6  sin  O.c 


1    sin  HIT 
J    n-  —  1 


op.       ^        /,./,■,„      o    sin  nir 


a,  =  ^Csin  nO  cos  6  sin  ^..7^  =  -  ■  ^^^,  and 

0 

1     mil  7>7r  I  7/ 

sin  «^  =  -  ;^.  V^     ^o(cos  ^)  +  5  -^^ -,  P^fcos  6) 

,   1    sin??7r  ToTi  /        m    1    -"^"~1^  75  /        m 

+  2-  ;7:r2"2  [_3^i(«os  6)  +  *  :^^r:z-^2 ^3(cos ^) 

If  11  is  a  whole  number  sin  //tt  =  0 ,  and  all  the  terms  of  (9)  vanish  except 
those  involving  P„_i(cos  6),  P„_^i(cos  6),  P„  +  3(cos  6)  &c.,  which  become  inde- 
terminate.    For  this  case  it  is  necessary  to  compute  a„_^  independently. 


Chap.  V.]  EXAMPLES.  183 

We  have 

2/?  —  ir 

«„_!  =  — - —  I  sin  n$F„ _j^(cos  6)  sin  d.dO 
2n  —  1 


-  r[cos  {n  —  1)^  — cos  (w  +  l)^]P„_,(cos  Q)^ 


le. 


TT  2?i — 1    1.3.5.  •••  ('2«,  —  3)  r-      A   i-   oo  /i\-i 

Hence  «„_ ,  =  -^-  .  0.4.6.  ■■  .(2,  -  2)  '^  ^''^  ^''-  '^^  ^^^^' 

and 


sm 


^^  1.3.->(2.-3)r 

4    2.4.--'(2»-2)L^  ^)^n-A^ 


„2 /.,  iNa 


+(2H-T)g:|;;;^;;ir;;::(;;+^);i^„<.os.)H----].(io) 


EXAMPLES. 
1.    Show  that 

csc^ 


=  J  [1  +  r>  (^)>.(eos  0)  4-  9  (^D'^^Ccos  ^)  +  13  (HI)"  A(cos  ^)  +  ■  •] 


whence 

1  TT 


v/f 


f  [1  +  5  @ V.(..)  +  9  (II)  A(.)  +  13  {^P.i^)  +  ■  •  •] 


[v.  Art.  90  (4)  and  Art.  82]. 
2.    Show  that 


ctn^ 
whence 


=  f  [3  (i)i>,(co.s  .)  +  7  (f)(^)p.(co.  .)  +  11  (|)(^)V,(cos  .,  +  .  ] 


Vl 


=  =  I  [^  a) A(..)  +  ^  (Da)''^.(^)  +  n  (|)0'a(.)  +  ■  ■] 


[v.  Art.  90  (4)  and  Art.  82]. 

3.    By  integrating  the  resnlt  of  Ex.  1  and  simplifying  by  the  aid  of  Art.  96 
(5),  obtain  the  development 

sin- ^  =  f  [3  (^) V,(..)  +  7  Q^JP,(^) 


184  ZONAL  HARMONICS.  [Akt.  97. 

whence         ^  =  J  rPo(cos  ^)  -  3  0-)  Pi  (cos  O)  —  !  (~yP^(cos  6) 

4.    By  integrating  the  result  of  Ex.  2  and  simplifying  by  the  aid  of  Art.  96 
(5)  obtain 


whence 
sin  e  = 


I  [i  P,(cos  ff)  -  5  (^)(i)  P,(cos  0)  -  9  (?)(ij)V.(eos  «)-•■•]. 


To  make  clearer  the  analogy  of  development  in  Zonal  Harmonic  Series  with 
development  in  Fourier's  Series  we  give  on  page  185  a  cut  representing  the 
first  seven  Surface  Zonal  Harmonics  Pi(cos  6),  P2(cos  6),  •  •  •P7(cos  0),  which 
are  of  course  somewhat  complicated  Trigonometric  curves  resembling  roughly 
cos^,  cos 2^,  •• -cos  7^;  and  on  page  186,  the  first  four  successive  approxi- 
mations to  the  Zonal  Harmonic  Series 

1  +  ^  P,(cos  6)-l.l  P3(cos  ^)  +  j|-  II  J'sCcos  6) [I] 

[v.  (1)  Art.  93],  and 

I  [Po(cos  0)  -  3  Q  Vi(cos  0)  -  7  (^)'^3(cos  0) 

-iK2sy^^(^°^^)--]M 

(v.  Ex.  3  Art.  97). 

7j-  7j- 

[i]  is  equal  to  1  from  ^  =  0  to  0  =  jj,  and  to  0  from   6  =  -^  to   0  =  7r;  and 

[ii]  is  equal  to  6  from  6  =  0  to  6---^=7r. 

The  figures  on  page  186  are  constructed  on  precisely  the  same  principle  as 
those  on  pages  63  and  64,  with  which  they  should  be  carefully  compared. 

98.  By  applying  Gauss's  Theorem  (B.  0.  Peirce,  Newt.  Pot.  Func.  §  31)  or 
the  special  Form  of  Green'' s  Theorem. 

(  (  f^^  '^dx  dij  dz  =  Cd,^  Vds  =  —  4.'7rCC  Cpdx  dy  dz , 


Chap.  V.] 


ZONAL    HARMONIC    CURVES, 


186 


ZONAL    HARMONICS. 


[Art.  9/ 


'-] 

"^ 

^^ 



^ 

~--,~' 

\ 

^\ 

,'— 

\-. 

»      / 

'>" 

<: 

0 

/ 

■^^    V 

.^. 

It 

__,. 

v_ 

V 


V.  page  184. 


Chap.  V.]  THIN    SPHERICAL    SHELL.  187 

[Peirce,  N.  P.  F.  §  49  (149)]  to  a  box  cut  from  an  intiuitely  thin  shell  of 
attracting  matter  by  a  tube  of  force  whose  end  is  an  element  of  the  surface  of 
the  shell  we  readily  obtain  the  important  result 

47rpK  =  I)J\-I),^r,.  (1) 

Avhere  p  is  the  density  and  k  the  thickness  of  the  shell,  Fi  the  value  of  the 
potential  function  due  to  the  shell  at  an  internal  point  and  V^  its  value  at  an 
external  point,  and  where  D„  is  the  partial  derivative  along  the  external  normal 
to  the  outer  surface  of  the  shell. 

If  we  have  to  deal  with  a  surface  distribution  of  matter  we  have  only  to 
replace  pK  in  (1)  by  O"  where  o"  is  the  surface  density,  whence 

47rcr  =  I)„  Fi  — 1)„  V^  '  (2) 

(v.  Peirce,  N.  P.  F.  §§  45,  46,  and  47). 

Formulas  (1)  and  (2)  enable  us  to  solve  problems  in  attraction  when  we 
know  the  density  of  the  attracting  mass,  and  problems  in  Statical  Electricity 
when  we  know  the  distribution  of  the  charge, by  methods  analogous  to  that  of 
Art.  94. 

For  example  let  us  find  the  value  of  the  potential  function  due  to  a  thin 
material  spherical  shell  of  density  p  and  radius  a. 

Since  V  must  be  a  solution  of  Laplace's  Equation  and  must  be  finite  both 
when    r  =  0    and    r  =  oo   we  have 


Fi  =2^^„r'«P,„(cos  6) 


Fi  and  Fg  must  approach  the  same  limiting  values  as  r  approaches  a.     Hence 

or  B„,  =  A,^/i:'"'  +  K 

D„  V,  =  D,  V,  =  -^{m  +  1)  -^—  P„,(cos  6) . 
Therefore  by  (1) 

47r/3K:  =^(2m  +  1)  J,X'~^^m(cos  0) 

if  K  is  the  thickness  of  the  shell. 


188  ZONAL   HARMONICS.  [Art.  99. 

Let  p  =/(cos  0)  =2)  C,„PJeos  6) 

1 
C„,  =  ^-^^^  jf{^)Pn.(x)d:r  by  Art.  90  (2). 


where 


and 
and 


Then  ^ttkC,,^  =  (2))i  +  l)A,„a"'-' ,  and 


^^-^4-«'^X2;7+i^"^'"^^'"^^^'  ^^^ 


Fe  =  ^-rraK^^;;^^  "^  P.(cos  ^)  .  (4) 


99.  We  can  noAv  get  the  value  of  the  potential  function  due  to  a  spherical 
shell  of  finite  thickness,  provided  that  its  density  can  be  expressed  as  a  sum  of 
terms  of  the  form    C>-^P„;(cos  6). 

Let  a  be  the  radius  of  the  outer  surface  and  b  be  the  radius  of  the  inner 
surface  of  the  shell. 

1st.  —  Let    p  ^  C?'^'P„,(cos  0).     Then  for  the  shell  of  radius  s  and  thickness  ds 

J\  =  4:7rsds  ^  ^'\_     ^1  P„,(cos  0)  by  (3)  Art.  98, 


Cs 


and  Fs  =  4:7rsds        '        ^— -j  P,„(cos  6)               by  (4)  Art.  98. 
Then  if    )•  <  h 

J  -      (2;«  +  l)        (A-  +  »^+3;            r'"  +  i                          ^^ 
/>  <  r  <  a 

J     -^J  '      2»i  +  lL(A;  +  m  +  3)?-"'  +  i 


if    ?•  >  a 


,jt-  —  m  +  2 


2d,  —  If   p  =^  C„,?'^'P,„(cos  ^)    the  solutions  will  consist  of  sums  of  terms  of 
the  forms  given  in  (1),  (2),  and  (3). 


Chap.  V.]  ZONAL    HARMONICS    OF    THE    SECOND  -KIND.  189 

EXAMPLES. 

1.  If  the  shell  is  homogeneous 

V^27rp(a'-b')    if    r<b, 

i  1        M 

V^-7rp{a^-P)-  =  ~     if     r>a, 

O  V  I' 

r  2h^        r-~l 

V=27rp\a'-j^--\      if     /j<r<a. 

2.  If  the   density  is  any  given  function  of  the  distance  from  the  centre 

F=—    if   r'>a,    and    F=  a  constant  if   r<ib. 
r 

3.  If  the  density  at  any  point  of  a  solid  sphere  is  proportional  to  the  square 
of  the  distance  from  a  diametral  plane 

a  \_r       i   r"^      ^         ^J 

4.  If  the  density  at  any  point  of  a  solid  sphere  is  proportional  to  its  distance 
from  a  diametral  plane 

if   r>  a.      Compare  Ex.  2  Art.  80. 

100.     We  have  seen  in  Art.  18  {e)  (3)  that 

^"(•-)  =  "'^"(-)/(l-.')tf„.(.r)]-'  (1) 

no  constant  term  being  understood  with     \  — ^^^   ,  ^■,,. 

1  J  (l-^')[-P.«(^)]' 

oN,--r.   .  N-io    is  a  rational  fraction  and  becomes  infinite  only  for   a;  =  1, 

(1  —  cc^)[P,„(a;)]^  '' 

cc  =  —  1,    and  for  the  roots  of    P,„(.r)=0,    all  of  which  are  real  and  lie 

between    —  1    and    1,    as    can    be    proved    by    the    aid    of     the    relation 

.  1      n^^-1)- 

If   .r-  >  1     I  — — -     is  finite  and  determinate  and  contains  no 

J  (l-.r-)[P,„(x-)]^ 

constant  term.     Hence  if   a;^>l 

for  the  constant  factor  of  (>,„(a:')  has  been  chosen  so  that    C  =  —  1 . 


190  ZONAL   HARMONICS.  [Art.  100. 

If   x^<il   the  second  member  of  (2)  is  not  finite  and  determinate,  and  we 
are  thrown  back  to  the  form  (1),  and  C  proves  to  be  unity. 

(1)  gives  us  readily 

^o(aO=|log^  (3) 

^i(^')=-l+^log^  (4) 

if    x^<l. 

(2)  gives  us  Q,(x-)  =  ^  log  |i|  (5) 

^,(.^)  =  -H-^log|^  (6) 

if   a'2>i. 

From  Art.  85  (10)  it  follows  that 

C  can  be  determined  and  is  equal  to     ^^ .  ^ '    if    a-^  <  1 ,    and  is  equal 

(-l)m2»^m!    .„      ,^,  ^^"'^• 

(2m)! 

Hence  ,„(.) .  (^ig^' ^,  [(.- 1)-/^,^,]  (T) 

if     x'<l, 

^    .     (-i)'«2'«?rt!  fZ"^  r^  2    iN»,  r      ^^-^      ~l  /«^ 

if     a;2>i_ 

(7)  and  (8)  give  us  for   Qo(x)  and  Qi(x)  the  values  already  written  in  (3), 
(4),  (5),  and  (6). 

By  the  repeated  application  of  the  formula 

(m  +  1)^,  +  i(a-)  -  (2m  +  l)xQ„,(x)  +  mQ^_,(x)  =0,  (9) 

which  may  be  obtained  for  the  case  where  x^<l  from  Art.  16  (13)  and  (14), 
and  for  the  case  where  cc^  >  1  from  Art.  16  (9),  any  Surface  Zonal  Harmonic 
of  the  Second  Kind  can  be  obtained  from  ^o(^)  and  Qi(x)  as  given  in  (3),  (4), 
(5),  and  (6). 


Chai'.  v.]  examples.  191 

Analogous  formulas  for  2)m{^)  ^^^^  1m(^)  can  be  obtained  without  difficulty 
from  Art.  16  (4)  and  (5).     They  are 

(m  +  l)^<z,„  +  i(.r)  -  (2ni  +  l)xp,,(x)  -  m'q,„_,(x)  =  0  (10) 

and  p^  +  ,(x)  -  {2m  +  l)q„,(x)  - p„,_,(x)  =  0  (11) 

and  they  hold  good  for  any  value  of  m. 

EXAMPLES. 

1.  Confirm  the  values  of  Qo(x)  and  Qi{x)  given  in  Art.  100  (3),  (4),  (5),  and 
(6)  by  expanding  them  and  comparing  them  with  Art.  16  (13),  (14),  and  (9). 

2.  If  the  value  of  V  on  the  surface  of  a  cone  of  revolution  can  be  expressed 
in  terms  of  whole  powers  positive  or  negative  of  r,  V  can  be  found  for  any 
point  in  space,  cf.  Art.  81. 

If      v=V(A„y'^-^^    when    6  =  a     then 

^\    '"      ^r'«  +  VP,„(cosa) 

3.  If      J^=2)6'^'«'''"  +  ".^)    ^^'^^^^^    0  =  a,    and    V=0   when   $  =  /3, 

T^_ V  {  I  r>-  4-  -^A  r^^».fcos  ^)P,„(cos  6)  -  F„,(co8  /3)()„,(cos  ^)-1 
-   Z^{^  'n      ^r'"-i;L^«(cosa)^7,„(cos^)-P„Xcos;8)(>,„(cosa)J  " 

4.  Find  V  for  points  corresponding  to  values  of  0  between  a  and  /3  when 
V  can  be  given  in  terms  of  whole  powers  of  r  for   6  =  a   and  for   0  =  /3. 

5.  Find  by  the  method  of  Art.  16  solutions  of  Legendre's  Equation  of  the 
form 

i(m  +  l)  ,^  _  -, ,  ^  (m-l)m(m-\-l)(m  +  2) 

22/2!)- 


. = ,pj.) = 1 + ;^^iv-^  (X - 1) + '"■  ^""v;  ";""'•  ^"^  (- - 1)-^ 


(,„  -  2)(m  -  l)m(m  +  1)(,»  +  2)(w  +  3)  ,^  _  -, , ,  , 

2^(31)'^  ^        ^  ' 

.  =  .,P„(.)  =  1  -  V^&^  (,  +  !)  +  (>"■  -  1)"'(- +  1)""  +  2)  (.  +  1). 

(M  -  2)(,« -!)«(»» +  l)(w  +  2)(m+ 3) 

2^^3  !)^  ^'^' """  ^    I" 

If  m  is  a  whole  number,  ^P,„{x)  =  P^^(x)  and  _iP,„(a;)  =  (— 1)'«P„,(^).  N,^ 
matter  what  the  value  of  m,  iF,„(x)  is  absolutely  convergent  for  —  1  <  »■  <  3 , 
and  _lP„^(x)  is  absolutely  convergent  for  —  3  <  x  <  1. 


19: 


ZONAL    HARMONICS. 


[Art.  100. 


6.    By  the  aid  of  (7)  Art.  16  show  that 

J'=  -^  sin  (n  log  r)k „(cos  6)  , 

Sir 

F^  -=  cos  (ii  log  r)A:„(cos  6)  , 
are  solutions  of  Laplace's  Equation 


F=  -r.  sin  (ii  log  ?')Z„(cos  0) , 

Sr 

r'=  -7;  cos  (11  log  /•)Z„(cos  6)  , 

V'r 


/•i>;(;-  TO  +  ^^Q  A(sin  ^  D,  F)  =  0 , 


A'„(a^) 


and 


^«(^)=-'Z-i  +  m(^)==-^-  + 


-+(i)\,[-+©i-+an 


,r«  + 


+  — X'  + 

^„(.r)  and  /„(a;)  are  convergent  if  0:-  <  1,  but  are  divergent  if   x^=l. 
7.    Show  by  the  aid  of  Example  0  that 


F^  -p  sin  (?i  log  ?')A''„(cos  6) . 
\r 

V=  -p  cos  (n  log  r)ir„(cos  6) . 


F=  -^  sin  (n  log  r)K„(—  cos  ^) , 

Sr 

F=  -^  cos  (w  log  r)K„(—  cos  ^) , 


are  solutions  of 


if  K„(x)  =  ,P_^,„,(x)  =  1 ^  (.r  -  1) 

,["-+a)i-'^©i  ^^, 

['"+©l"-+(Dl-'+@'] 

23(3  !)2 


(^_1)3_^... 


Chap.  V.]  EXAMPLES.  193 

and 

K„(-  x)  =  _,P_^_,  „,(.t)  =  1  + ^  i^x  +  1) 

,[-+©;][;:H-©l, ,.,, 

-ff'„(cos  ^)  is  convergent  except  for    0  =  '7r,    and    7t„(— cos^)     is  convergent 
except  for    6^=0. 

^n(^))  4(^0 J  -^'wC-'^))  and  7i'„(— .>■)  are  sometimes  called  Conal  Harmonics. 
They  are  particular  values  of  z  which  satisfy  Legendre's  Equation  written  in 
the  form 


0. 


For  an  elaborate  treatment  of  them  see  E.  W.  Hobson  on  "A  Class  of  Spherical 
Harmonics  of  Complex  Degree."     Trans.  Camb.  Phil.  Soc,  Vol.  XIV. 

8.  If    V=S\r)    when    e  =  j3, 

9.  If    V=f{r)    when   6  =  jB   and   r<a,    and    V={)   when   r:=a, 

—      ^  a 

r=-J-  Cd\  Ce  if(ae>^)  ^."^^'^^  f^  sin  aX  sin  fa  log  -)da  ;     if     ^  <  /3 . 

TT    ^?'J         J         ^         ^/i„(C0S/3)  V  "^  O/  '  ^ 

— «  0 

10.  If    V  =  /(>')    when    ^  =  /3   and   a<)'<b,    and    r=()   when   r  =  a 
and  when   rz=b, 

y       7i;„-(cos  ^)    .     rw7r(logr  — loga,)~| 
^^£/^^  '"  A'„,.(cos  y8)  ^"'  L      log  h  -  log  «     J 

where  m'  = :; ^^^:!i^ and 

log  b  —  log  a 

r ""« 


'"       log  6  —  log  a  ^  rj     -'^^      '         log  b  —  log  a 


194  ZO^^AL    HARMONICS. 

11.  If   6>  fi   COS  6  must  be  replaced  by  (—  cos  6)  in  examples  8,  9,  and  10. 

12.  If    r =/(>•)    when    6  =  ^,    and    r=0    when   e  =  y, 

-,.  1       /»  7.    r   ^  y./   i\  ^"afCOS  6)IJC0S  y)  —  /^„(COS  7)4(C0S  ^)  r    /\         1  n-i  7 

ttV/^J      J    -^  ^    ^  A;a(cos  /3)4(cos  y)  —  A;„(cos  y)4(cos  ^)        '-  ^  s   yj     , 

— ^      0 

if    /3<^<y. 

13.  If    r =/()')    when    ^  =  /3    and    a<r<b,    V=0    when    ^  =  y    and 
a<i  )'  <Cb,    and    F=  0    when    ;•  =  a    and  when    r^b, 

j;r_^sf,    A-m'(cos  ^)<^)»'(cos  y)  —  A-,„'(cos  y)Z,„'(cos  ^)    .     m7r(log  r  —  log  a) 
~-^    "'  k,n'(GOS  (3)lm(G0Sy)  — k,n'{COSy)l,„-(COS  (3)  log  6  —  log  a 

where  ?«'  =  ^ — and 

log  b  —  log  a 

b 

^"'  =  log  6- log  «  \'r  J  ^^•^(^'^')  ^^^  log  i- log  «  ^^  5 

0 

if   (3  <0<y    and    «<;■<//. 

14.  If    V=f(r)    when    ^  = /3    and    (;<r<i,    and    T'=^    when    r=  a 
and    X>,  J^' -{-  hV=0    when    /•  =  i , 

'"="      /!„  (cos^)  .  ^,. 

^=S^m  ^^'^'t m  sin  (  a,„  log    )  ,  where 

^      /f,,^^  (cos  /3)        V  ''  / 

log  - 

^^ ""  a,r:(iog  b  -  log  «) V/i^'L^Hiog  ^  -  log  '0  + 1]/"'^-^^"'"^ ''''  '"'"''•'^^ 

and  a,„  is  a  root  of  the  equation 

a  cos  (a  log  A  +  hb  sin  (^a  log  -)  =  0         v.  Art.  68  Ex.  5. 


CHAPTER    YI. 


SPHERICAL    HARMONICS. 


101.  When  we  are  dealing  with  problems  in  finding  the  potential  function 
due  to  forces  which  have  not  circular  symmetry  *  about  an  axis  and  are  using 
Spherical  Coordinates,  we  have  to  solve  Laplace's  Equation  in  the  form 

rD:-{r V)  +  J-,  i»e(sin  0  DeV)  +  -Ar^  I>1  ? '  =  < »  (1) 

[v.  (xiii)  Art.  1]. 

To  get  a  particular  solution  of  (1)  Ave  shall  assume  as  usual  that  V  is  a 
product  of  functions  each  of  which  involves  but  a  single  variable. 

Let  V-^=  i?.©.$;  where  B,  involves  r  only,  ©  involves  Q  only,  and  O  ^  only. 
Substitute  in  (1)  and  we  get 

r<nrR)  1        4"^^  I)  1        d^^ 

R     d)-'     "^©sin^  dO  '^^sm'ed4-  ^^^ 

,/,.     ^d®\ 
r  sin^  6  d:\rR)       sinO    v"      f/^/  __ld^ 
B         r/r^     ^0  dd  ~       <Dr/<^-" 

As  the  first  member  does  not  contain  ^  the  second  member  cannot  contain 
^,  and  as  it  contains  no  other  variable  it  must  be  constant;  call  it  n^.  Equa- 
tion (2)  is  then  equivalent  to  the  two  equations 

^^)  +  ^fc!ll-^=0  (4) 

R     dr"'     ^  ©  sin  d  dO  sin^  0  ^  ^ 

(3)  has  been  solved  before  and  gives  us 

(^^  A  cos  «<^  +  B  sin  nx^  (5) 

[v.  Art.  13(a)]. 

The  first  term  of  (4)  does  not  involve  6  and  the  second  and  third  terms  do 
not  involve  n 

*  See  note,  page  12. 


and 


196  SPHERICAL   HARMONICS.  [Art.  101. 

—     ^  .^       must,  then,  be  a  constant;  we  shall  call  it    iii(m-{-l)    as  in  Art. 
13(c).     Then  (4)  breaks  np  into 

r  ^^^^  =  m{m  +  1)R  (6) 

— .  m  +  [-(-  +  1)  -  -^>  =  0  .  (7) 

sm  B  (IB  '    [_    V      '     /      gjj-,2  ^  J  V  / 

(6)  was  solved  in  Art.  13(c)  and  gives 

R  =  A,r"'  +  B,r-"^-\  (8) 

If  in  (7)  Ave  replace  cos  ^  by  /*  we  get 

ib'-  "'^  |]  +  ["•('" + 1)  -  r^G® = » '  w 

the  equivalent  of 

(1  -  ^')  S  -  -■'^  I + [•»('» + 1)  -  r^J  ■' = "         (i»> 

[v.  (17)  Art.  85],  which  was  solved  in  Art.  85  for  the  case  where  m  and  n  are 
positive  integers  and   w  <  m  +  1.     v.  (18)  and  (19)  Art.  85. 
From  (19)  Art.  85  we  get  as  a  particular  solution  of  (9) 

if  we  restrict  ourselves  to  whole  positive  values  of  m  and  n,  as  we  shall  do 
hereafter  unless  the  contrary  is  explicitly  stated,  and  suppose  m  not  less 
than  n. 

A  second  but  less  useful  particular  solution  of  (9)  is 

Combining  our  results  we  have  as  important  particular  solutions  of  (1) 

V=  r"'(A  cos  ncl>-\-B  sin  »<^)  sin"  0  ^^^^^  '  (12) 

and  V=  ^  (A  cos  n^  +  B  sin  nxf>)  sin"  0  ^^"^'"[^^  '  (13) 

where  m  and  n  are  positive  integers  and   n  <  /n  +  1- 


Chap.  VI.]  TESSERAL   HARMONICS.  197 

(P^P  (u)  d"P  (a) 

102.     sin"  6 r^—     or     (1  —  ix% r^—      is  a  new  function  of  a,  that  is 

d/x"  ^  dfi" 

of  cos  $,  and  we  shall  represent  it  by  F„''(ijI')  *  and  shall  call  it  an  associated 

/miction  of   the  nth.  order  and  mth  degree.      It  is  a  value  of  ®  satisfying 

equation  (9)  Art  101. 

By  differentiating  the  value  of  P„,(x)  given  in  (9)  Art.  74  we  get  the  formula 

Pm  (/^)  -  2.  ^, ,  (,,,  _  ,,) !  L^  2.  (2m  -  1)  ^ 

(»^  -  70(»^-  -  n  -  l)(m  -  n  -  2)(m-?^-3)  _, "I  .;l) 

■^  2.4.(2/«-1)(2;m-3)  ^  J^^ 

the  expression  in  the  parenthesis  ending  with  the  term  involving  x^  if  m  —  n  is 
even  and  with  the  term   involving  x  if  vi  —  n   is  odd. 

Eor  convenience  of  reference  we  give  on  the  next  page  a  table  from  which 
P,;^(fi)  can  be  readily  obtained  for  values  of  m  and  n  from  1  to  8. 

cos  ?i<^Pj(/i)    and    sin  n(f>  Pj}(fi),    that  is, 

.      ^d"PJa)          ,       .            .      ^d"Pjfi) 
cos  ncf>  sm"  6  — r^ — ■     and    sm  neb  sm"  t* r-— 

are  called  Tesseml  Harmonics  of  the  mih  degree  and  nth  order,  and  are 
values  of  V  which  satisfy  the  equation 

m,Qu  +  1)  r+  ^^  Deisms  A  V)  +  ^  A,^  V=  0  (2) 

or  its  equivalent 

m(m  + 1)  r+  zv[(i  -  ^^)i),  r]  +  ^^,  i>|F=  0  .  (3) 

There  are  obviously  2m  +  1  Tesseral  Harmonics  of  the  mth  degree,  namely 
PM,      cos<^sin^^i^\  sin  <^  sin  ^^^^^ 

cos2c^sin^/^^^,  sin2<Asin^^^^^ 

cos  7»  (^  sin-  0  '^"'f"^^-* ,  sin  7hc/,  sin'"  ^  '^"'f"'f^-*-  • 

If  each  of  these  is  multiplied  by  a  constant  and  their  sum  taken,  this  sum 
is  called  a  Surface  Spherical  Harmonic  of  the  7>?.th  degree,  and  is  a  solution  of 
equations  (2)  and  (3).     We  shall  represent  it  by  Y„^(fl,  </>)  or  by  Y„^(0,  (t>). 

*  Most  of  the  English  writers  represent  this  function  by  T',;^^). 


198 


SPHERICAL    HARMONICS. 


[Art,  102. 
Table  'for 


- 

m 
1 
2 
3 
4 
5 
6 
7 
8 

n=.l. 

,1  =  2. 

1 

n  =  3. 

1 

3/. 

3 

|(5m-'-1) 

15m                          !                     15 

|(7.-3.) 

f(7M-l) 

105m 

^  (21m^  -  IV-  +  1) 

105  ,.,  3        , 

f(..-.) 

"^  (33m5  -  30m^^  +  5m) 

^  (33m*  -  IBm^  +  1) 

315 

-  (42V  —  495m-'  +  135m2  —  5) 

'-^  (143m5  -  110m«  +  15m) 

o 

^  (143m*  -  06m2  +  3) 

^  (715m'  -  IOOIm^  +  385m3  -  35m) 

^(mM^-HSM-i  +  SSM^-l) 

^  (39m5  -  26m-''  +  3m) 

r '"!",„ (/Lt,  <^)    and    — —^  l^m(y"'j  ^)    ^^'S  called  *SoZiVZ  Spherical  Harmonics  of  the 

mtli  degree,  and  are  solutions  of  Laplace's  Equation  (1)  Art.  101. 
To  formulate:  — 

r„,(M,  «/>)  ^Xf'^'^"  ''''^  ""^  '^'''"  ^  '^""^"'f^^  +  B„  sin  «<^  sin"  ^  '^""^'"f^H    (4) 

n  =  0 

or  r,,(/i,  <^)  =  A,P,,{ix)  +^IA,  cos  n^P^lfi)  +  B„  sin  7^<^P;,;(/^)]  (5) 

is  a  Surface  Sijlierical  Harmonic  of  the  mth  degree. 

A  Tesseral  Harmonic  is  a  special  case  of  a  Surface  Spherical  Harmonic,  and 
a  Zonal  Harmonic  a  special  case  of  a  Tesseral  Harmonic;  -P„j(/i)  being  the 
Tesseral  Harmonic  of  the  zeroth  order  and  the  mth  degree;  it  might  be 
written    P,^(/i). 

EXAMPLES. 

1.    Show  that 

reduces  to 

(1  -  X')  g  -  2(M  +  1)^  ^  +  Vn(m  +  1)  -  n(n  +  1)]//  =  0 

if  we  substitute    (1  —  x-)\y   for  z,  even  when  m  and  n  are  unrestricted. 


Chap.  VI.] 


TABLE   FOE    ASSOCIATED    FUNCTIONS. 


199 


csc«^P„»  =  ^^>. 


n  =  4. 

11  —  5.' 

n  =  0. 

71  =  7. 

n  =  8. 

105 

94.5m 

945 

f  (11.-1) 

10395/x 

10395 

^(13^3-3,) 

'T<''^  » 

1.35135/x 

1.35135 

l-f^\65.4_  26.^  +  1) 

''Ti'^'  -) 

'Td^^   1) 

2027025m 

2027025 

2.    Show  that  if  in  the  second  equation  of  Ex.  1  we  let     y  =  2  a^x''   we  get 
(»?,  —  71  —  k)Om  +  n-\-l  +  k)  .       .    .    ^ ^. 

whence   z=2yZ(a:)   and  z  —  qZ(^)   are  sohitions  of  the  first  equation  of  Ex.  1, 
no  matter  what  the  values  of  m  and  n,  if 

i.;;W  =  (1  -  .■^)S  [i  -  ('—")('-  + -  +  r> ,, 


and 


"T-  41  .T        •■•J 

(v/A  —  w  —  1)  (m  —  ?^  —  3)  (m  +  7?,  +  2)  (m  +  ».  +  4)    g_      .-| 
^  ^  5!  -^^        "J- 

If   m.  —  w   is  a  positive  integer,  2j>",(x)  or  2',"(a;)  will  terminate  with  the  term 
involving  x"'~";  and  in  that  case 


(ill  —  n)(iii  —  n  —  1)    „j_^  _ 
2.(2/«-l)  ^"" 


,    {m  —  11)  (in  —  11  —  1)  (m  —  n  —  2)  (>/<-  —  ?<,  —  3) 
"*"  2.4.  (2»i  —  1)  (2m  —  3) 


^,„_„_4 "I   ^ 


liUU  SPHERICAL    HARMONICS.  [Art.  103. 

the  parenthesis  ending  with  a  term  involving  x°  if  ui  —  n  is  even  and  x  if 
/n  —  n  is  odd,  is  a  solution  of  the  first  equation  of  Ex.  1.     If  m  and  ti  are 

integers  this  value  or  z  is  -^^ — -; — —  -r'[(^)  . 

(2m)  I  ^ 

103.  We  have  seen  in  the  last  chapter  that  in  many  problems  it  is  import- 
ant to  be  able  to  express  a  given  function  of  cos  0,  that  is  of  /x,  in  terms  of 
Zonal  Harmonics  of  fi.  So  it  is  often  desirable  to  express  a  given  function  of 
fi  and  <ji  in  terms  of  Tesseral  Harmonics  of  fi  and  eft. 

If,  for  example,  we  are  trying  to  find  the  Potential  Function  due  to  certain 
forces  and  have  the  value  of  the  function  given  for  some  given  value  of  r, 
that  is,  on  the  surface  of  some  given  sphere  whose  centre  is  at  the  origin  of 
coordinates,  of  course  the  given  value  will  be  a  function  of  6  and  <^  and  if  we 
can  express  it  in  terms  of  Spherical  Harmonics  of  6  and  ^  we  have  only  to 
multiply  each  term  by  the  proper  power  of  r  to  get  the  required  solution  of 
the  problem.  For  we  shall  then  have  a  value  of  V  satisfying  Laplace's 
Equation  and  reducing  to  the  given  function  of  6  and  ^  on  the  surface  of  the 
given  sphere. 

104.  Suppose  that  we  have  a  function  of  ft  and  ^  given  for  all  points  on 
the  unit  sphere,  that  is,  for  all  values  of  /i  from  —  1  to  1  and  for  all  values  of 
<^  from  0  to  2'TT,  fi  and  <f>  being  independent  variables,  and  that  we  wish  to 
express  it  in  terms  of  Surface  Spherical  Harmonics. 

Assume  that 

f(H;  ^)  ^X[_A,nPM  +X(^i.,„  cos  ncl>P:(fji)  +  5„,,„  sin  »c^P„X/i))]  •    (1) 

Let  us  consider  first  a  finite  case,  and  attempt  to  determine  the  coetticients 
so  that 

/>,  <!>)  =X[^°''«^"'^^^  +X(-^".'«  ««s  ''<t>P:>M  +  ^n,n.  sin  ?i<^P:(;u))  J    (2) 

shall  hold  good  at  as  many  points  of  the  sphere  as  possible.  The  expression 
in  brackets  in  the  second  member  of  (2)  is  a  Surface  Spherical  Harmonic  of 
the  mth  degree  and  contains  2/??  + 1  constant  coefficients.  The  whole  number 
of  coefficients  to  be  determined  is  then  the  sum  of  an  Arithmetical  Progression 
of  i^  +  1  terms  the  first  term  of  which  is  1  and  the  last  is  2^?  +  1 ,  and  is 
therefore  equal  to  (p  +  1)1 

Let  the  interval  from  yu,^  —  ltofi=:lbe  divided  into  ])  -{-2  parts  each  of 
which  is  Afi  so  that  (j)  +  2)A/i  =  2,  and  let  the  interval  from  <^  i=  0  to  <^  =  27r 
be  divided  into  p-\-2  parts  each  of  which  is  A<^  so  that  (p  +  2)A^  ==  27r. 


Chap.  VI.]         DEVELOPMENT    IN    SPHERICAL    HAIIMONIC    SERIES.  201 

Then  if  we  substitute  in  equation  (2)  in  turn  the  values  ( —  1  -|-  A//.,  ^4>), 
(-  1  +  2A/X,  A<^),  •  •  •  [-  1  +  (/>  +  l)Afi,  Ac^];  (-  1  +  A/i,  2Acjf>), 
(-  1  +  2\fi,  2AcA),  •••[-!  +  iP  +  1 )  Ay^,  2A<^];  •  .  •  [-  1  +  A/.,  (p  +  1)A<^], 
[-  1  +  2A/ti,  (p  +  l)Ac/>),  •  •  •  [-  1  +  (^>  +  1)A/^,  (p  +  1)A<^];  since  the  first 
member  in  each  case  will  be  known  we  shall  have  Qj  -\-  1)^  equations  of  the 
first  degree  containing  no  unknown  except  the  (^v  +  1)"^  coefficients,  and  from 
them  the  coefficients  can  be  determined.  When  they  are  substituted  in  equa- 
tion (2)  it  will  hold  good  at  the  (p  +  1)^  points  of  the  unit  sphere  where  p-{-l 
circles  of  latitude  whose  planes  are  equidistant  intersect  ^>  + 1  meridians 
which  divide  the  equator  into  equal  arcs.  If  now  p  is  indefinitely  increased 
the  limiting  values  of  the  coefficients  will  be  the  coefficients  in  equation  (1), 
and  (1)  will  hold  good  all  over  the  surface  of  the  unit  sphere. 

To  determine  any  particular  constant  we  multiply  each  of  our  (/;  + 1)^ 
equations  by  A/tt  A<^  times  the  coefficient  of  the  constant  in  question  in  that 
equation  and  add  the  equations  and  then  investigate  the  limiting  form 
approached  by  the  resulting  equation  as  /^  is  indefinitely  increased. 

As  p  is  indefinitely  increased  the  summation  in  question  will  approach  an 
integration;  and  since  d/jbd4>^=  —  sin  O.dO d(f>  is  the  element  of  surface  of  the 
unit  sphere,  and  as  the  limits  —  1  and  1  of  yu,  correspond  to  tt  and  0  of  0  the 
integration  is  a  sm^face  integration  over  the  surface  of  the  unit  sphere. 

In  determining  any  coefficient  as  A^„^  in  (1)  the  first  member  of  the  limiting 
form  of  our  resulting  equation  will  be 

p7</)  (/(/"■,  4>)  COS  n(^  P;^,(/x)d/j,. 

In  the  second  member  we  shall  come  across  terms  of  the  forms 

Cdcf)  Tsin  Icj)  cos  ncji  P„[(iii)P,][(fi)dfx,,    Cd(f>  Tcos  left  cos  ncl>  P,l{^^)Pl{p)dfjL, 

0-1  0-1 

2t  1  2T  1 

Cd(i>  Tsin  «<^  cos  ??</>  [P,;;(/A)]'^r//ti,    p/</)  fcos-  n<^  lP:^^{^l)Jd^l, 
and  other  terms  all  of  which  come  under  the  form 

<fdci,^Y,{^l,<^)Y,,{^l,<iy)d^l, 

(I        -1 
where    Y^{^i,  (j>)   and    F,(At,  <^)   are   Surface   Spherical   Harmonics  of  different 
degrees. 

If  we  are  determining  a  coefficient  B„,„  the  only  difference  is  that  sin  n(f> 
and  cos  n(j)  will  be  interchanged  in  the  forms  just  specified. 


202  SPHERICAL    HARMONICS.  [Akt.  105. 

105.      The  integral  over  the  surface  of  the  unit  sphere  of  the  product  of  two 
Surface  Spherical  Harmonics  of  different  degrees  is  zero. 

That  is  lfd<i>j  Y,{fi,  </>)  Y„,{fji,  4>)dp,  =  0 .  (1) 

0  -1 

For  as  we  have  seen    V'^=)^Yj(/j,,  </>)    and     F:=  ?-'"y,„(yLi,  (f>)    are  solutions  of 
Laplace's  Equation.     Hence  by  Greenes  Theorem 


C{UD„  V—  VD„  Zr)ds  =  0 


V.  Art.  92. 


J}^^  U  =D,.U=  Ir' - 1  Yi(fi,  <l>)  ; 
UD„  V-  VD,,  U=  („>  -  ly-^'"-^  Y,(f.,  </.)  Yjfji,  c^), 
=  (>H-l)Y,(fx,ct>)Y.„^(fi,ct>) 
on  the  surface  of  the  unit  sphere;  and 

(m  -  l)j^Y,(fi,  (/>)  !;„(/.,  <f>)ds  =  (m  ~  Ij^dcfyJ Y,(fx,  <^)  Y^(fM,  <j>)dfi,  =  0 

0  —1 

277  1 

Jrf<^J'F,(/x,</>)Y-,„(/.,c^)f7/. 


Hence  unless  /  =  ?h 

277  1 

0 


EXAMPLES. 

1.  Obtain  (1)  Art.  105  directly  from  the  equation 

m(m  +  1)  Y,^(fi,  </,)  +  D^[(l  -  /x'^)Z)^  Y^(fi,  «^)]  +  -^,  Dl  Y^(fM,  <j>)  =  0 
V.  (3)  Art.  102,  and  Art.  91. 

2.  Show  that  the  integral  over  the  surface  of  the  unit  sphere  of  the  product 
of  two  Tesseral  Harmonics  of  the  same  degree  but  of  different  orders  is  zero. 

Suggestion  : 

27r  2t  2ir 

j  sin  k<p  COS  l<f).dcf>  =  j  sin  l-cf>  sin  lcf>.d(f)  =  j  cos  kef}  cos  Iffi.dcj)  :=  0  . 


106. 


CFp(fi)FZ(luL)dfi  =  0     unless     I  = 


(m±nll     .^     ^_ 


2m  +  1  (m.  —  n) ! 


Chap.  VI.]       DEVELOPMENT    IN    SPHERICAL    HARMONIC    SERIES.  203 

For 

by  integration  by  parts. 

Replacing   n   by   n  —  1    in    equation    (2)    Art.   84    and    remembering    that 

—    ^^"'j'         is  a  possible  value  of  s<«-i'  we  get 

(1  - .')  '^9;i#-^  -  -"«.  ^ + [«(». + 1)  - «(« - 1)]  '^^ = 0, 

or  if  we  multiply  by  (1  —  /x-)"~'^ 

+  ^;,,  +  n)^ni  -  „  +  1)(1  -  /i^)«-^  '^"  /f!'{^^  =  0, 

(111 
or 

Hence  follows  the  reduction  for  mala 

—  1 

Using  this  formula  n  times  we  get 

-i  •  -1 

=■-  0     unless     /  ^  /». 

2        (w+».)!     .„     , 

=  7, ]— T  7 t"i     ^^     l  =  tn 

2m  -\-l  {m  —  m)! 

V.  Art.  89  (4)  and  (5). 


204  SPHERICAL   HARMONICS.  [Art.  107. 

107.  We  are  now  able  to  complete  the  solution  of  the  problem  in  Art.  104 
and  since  j  cos^  nt^t-dc^  =  j  sin^  n4>.d<j>  =  tt  and  |  d(^  =  27r  we  get  as  the 
coefficients  in  (1)  Art.  104 

An,  =  ^-^^P^ffifl,  <l^)PUl^)dfl  ,  (1) 


_  2m  +  1    (m  —  n) ! 


2m  -\-  1    (m  —  n) 

^n,™  -      27r      ■  (7H  -i^~ 


2^  1 

'^.Jd^pit^,  <A)  sin  n^P'^{fi)dix ,  (3) 


whence 


and  the  development  holds  good  for  all  values  of  /*  and  </>  corresponding  to 
points  on  the  unit  sphere,  provided  only  that  the  given  function  satisfies  the 
conditions  that  would  have  to  be  satisfied  if  it  were  to  be  developed  into  a 
Fourier's  Series. 

If  we  use  fXi  and  c^i  in  place  of  fi  and  cf>  in  (1),  (2),  and  (3),  we  can  write  (4) 
in  the  form 


+X(^^!/'^^^/^^'^^ '  <^i)^."(/^)^".(/^0  COS  n(cf>  -  cl>,)df,,'j  .    (5) 

Formulas  (1),  (2),  (3),  and  (4)  are  convenient  for  actual  work;  (5)  is  rather 
more  compactly  written. 

108.     As  an  example  let  us  express    sin^  0  cos^  0  sin  <f)  cos  <^    in  terms   of 
Surface  Spherical  Harmonics. 

Here  f(/x,  </>)  =  -  ,x\l  —  fx^)  sin  2<f> . 


_2jn  +  l 
A--      Sir 


jli\l  -  fi')P,,(fi)dfiPm  2^.d4>  =  0 , 


Chap.  VI.]  ILLUSTRATIVE    EXAMPLE.  205 


A,,n  = 


2w.  -\- 1    (vi  —  n) ! 


47r 


im  +  n)\j  ^'^^  ~  l^')Pl{l^)dfxj  sin  2<^  cos  n<i>.d<i>  =  0  ., 


5„„„  =  —^  ■  ^^^;  J /^-^(l  -  A^^)P;,;(/.)r//. Jsin  2<^  sin  .</..rf</. , 
=  0     unless     ??=  2 . 

Sir  2n- 

If   71  ^  2  I  sin  2</»  sin  vcfi.dcf)  =  (  sin^  2</).r/^  =  tt,  and 

—  1 

—  1  —1 

by  repeated  integration  by  parts, 

=  0     if     w.>4, 

1 
=  720  C{iJi,'  -  lydfi  =  '^'^^^     if     /M  =  4  , 

-^  ^ 

d  _  J_   9   2J   4096  _    1 

^^  '•■'~2*4!'4'6!'     7     ~  105 ' 

By  a  like  process  we  find 

^23  =  0     and     B.o^—-  Hence 

42 

sin^  ^  cos^  0  sin  <^  cos  c^  =  —  P.|(/Lt)  sin  2<^  +  -—  P|(/ti)  sin  2<^ ,  (1) 

4_  lOo 

=  —  sin-  ^  sin  2c^  +  :q  sin'^  ^  (7yLi2  —  l)  sin  2<^ .  (3) 

The    required    expression    might    have    been    obtained    without   using   the 
formulas  of  Art.  107,  by  a  very  simple  device,  as  follows: 

sin^  6  cos^  6  sin  0  cos  <^  =  -  /i-  sin-  d  sin  2<j!) .  (4) 


206  SPHERICAL   HARMONICS.  [Art.  109. 

If  now  we  can  express  fi"^  in  the  form    V — —f-^    the  work  will  be  done. 

'^  4.3      f/yLt^    ' 

^'  =  i  ^*(/^)  +  \  ^2(/^)  +  \  Po(f^) ,  (5)  Art.  95. 


whence  ^2  _  _  __i^/ _^  _      _^ 


2_  d'PM      2.  ^^'A(/^) 
105     rf/x-^     ~^21     t^/i^ 


and  substituting  this  value  in  (4)  we  get  (2). 

EXAMPLES. 


1.    Show  that 

cos«  e  sin^  6  sin  <^  cos^  <^  =  [^  P,f (^)  +  ^  P.^/i)]  sin  3<f> 


2.    Show  that 
cos 


r  5  9  2'  134'  "1 

2c/>  =  2  cos  2c^  [-  Piif.)  +  -^-  P|(/.)  +  -gy--  P|(/x)  +  •••]• 

3.  If  in  a  problem  on  the  Potential  Function  F=/(/i,  <^)  when  ?■  =  «,  we 
shall  obviously  have 

^=X^*  [A,«^™(/^)  +X(A,n  COS  .^<^  +  P„,^  sin  ncf>)P;:,(,u,)~] 

m  ^  0  n  =  1  -" 

at  an  internal  point  and 

^^i)^'  \^ArnP,M  +X(A,,n  COS  ^C^  +  P„,„,  Sin  ?^«^)P»(/x)] 

at  an  external  point,  where  Aq,„,  A„„^,  and  B„„^  have  the  values  given  in  (1), 
(2),  and  (3)  Art.  107. 

4.  Solve  problems  (3),  (4),  and  (5)  of  Art.  94  for  the  case  where  Fis  not 
symmetrical  with  respect  to  an  axis. 

109.  Any  Solid  Spherical  Harmonic  r'"I'„(yu.,  ^)  being  a  value  of  F  that 
satisfies  Laplace's  Equation  in  Spherical  Coordinates  will  transform  into  a 
function  of  x,  y,  and  z  satisfying  V^F^O  if  we  change  to  a  set  of  rectangular 


Chap.  VI.]    ANOTHER    DEFINITION   OF  A  SPHERICAL    HARMONIC.  207 

axes  having  the  same  origin  and  the  same  axis  of  X  as  the  polar  system. 
Moreover  the  new  function  will  be  a  homogeneous  rational  integral  Algebraic 
function  of  a;,  y,  z,  of  the  mth  degree. 

For  each  term  of  r"*  cos  w^P;/,(/x)  is  of  the  form 

Qj,m  cos»--^<^  sin-^'<^  sin"  6  cos'"--'-"  0 

where  2k  <n-\-l     and     21  <  m  -  n  +  1. 

This  may  be  written 

Cr'^K  r"'-2'-»  cos'"--'-"  6.  r"-^^  sin"-^*-  $  cos"--*  <^.  ?-*  sin  2*'  0  sin^*'  <^ 

which  becomes  C\x^  +  i/-  +  ,t;^)'x'"--'^"  t/"--^'s-^, 

and  is  a  homogeneous  rational  integral  Algebraic  function  of  x,  y,  and  z  of  the 
mth  degree.  The  same  thing  may  be  shown  of  each  term  of  r'«  sin  n<f)P;'(^). 
Consequently  r'"^  Y„^(im,  ({>)  is  a  homogeneous  rational  integral  Algebraic  func- 
tion of  the  mth  degree  in  x,  y,  and  ,~. 

110.  Any  homogeneous  rational  integral  Algebraic  function  S.^(x,  y,  z)  of 
the  mth  degree  in  x,  y,  and  2;,  which  is  a  value  of  V  satisfying  V^F=0  con- 
tains 2))i-\-l  arbitrary  constant  coefficients. 

For   S„^(x,  y,  z)  will  in  general  consist  of   ^ '-^ ■ — ^  terms  and  will 

therefore  contain  ^ -^— —   coefficients. 

V^'?,„(.r,  y,  z)  will  be  homogeneous  of  the  (m.  —  2)d  degree  and  will  contain 

^ ~  coefficients,  Avhich,  of  course,  Avill  be  functions  of  the  coefficients  in 

S,„(x,  y,  z).      Since   V-'S'„,(.7',  y,  z)  =  0   independently  of  the  numerical  values 

of  X,  y,  and  z  the  .-, coefficients  in  \7'^S,,^(x.,  y,  z)  must   be   separately 

zero,  and  that  fact  will  give  us  — —^ equations  of  condition  between  the 

^^ -^~ — '-^  original  coefficients  and  will  leave  ^^ ^ — '^^—^ ~ — '--_ '- 

or  2m.  -}-  1  of  them  undetermined.  S.„^(x,  y,  z)  contains,  then,  the  same  number 
of  arbitrary  coefficients  as  r"^Y„^(iJ,,<}>). 

We  can  then  choose  the  coefficients  in  r'"  F,„(/Lt,  </>)  so  that  it  will  transform 
into  any  given  Sjj^(x,  y,  z). 

Consequently  a  Solid  Spherical  Harmonic  of  the  m\X\  degree  might  be 
defined  as  a  homogeneous  rational  mtegral  Algebraic  function  of  x,  y,  and  z, 
S,^(x,  y,  z),  of  the  mth  degree  satisfying  the  equation  \^'^S„^(x,  y,  z)  =0;  and  a 
Surface  Spherical  Harmonic  of  the  mth  degree  as  such  a  function  divided  by 
(a-2  +  y2  _^  z')'^,  that  is  by  r"K 


208  SPHERICAL   HARMONICS.  [Art.  111. 

EXAMPLES. 

1.  Show  that  if  S„^(.^•,  y,  z)  is  a  Solid  Spherical  Harmonic  of  the  mth  degree 

V^[r".%„(.r,  7/,  -t)]  =  n(2m  +  n  +  1)/--^S;„(^,  y,  ,t;). 
Suggestion : 
V'S„,  =  0.      VV  =  ^-      D,S^  =  '"f^-      (I),ry+(D^ry+(B,7f  =  l. 

2.  Show  that  if  f„(x,  y,  z)  is  a  rational  integral  homogeneous  function  of  x, 
y.  and  z  of  the  ?;th  degree  it  can  be  expressed  in  the  form 

f„{x,  y,  z)  =  S„{x,  y,  z)  +  r\S,,_,(x,  y,  z)  +  7-'S„_,{x,  y,  z)  +  ■  ■  ■ ,  (1) 

terminating  with  r"-^S'i(.T,  y,  z)  if  n  is  odd,  and  with  7^''So(x,  y,  z)  if  n  is  even. 

Suggestion:  If  a  term  rS,^_-^  were  present  in  the  second  member  of  (1),  and 
we  were  to  operate  with  V^  on  both  members  we  should  by  Ex.  1  have  a  term 

—  '?„_!  which  would  be  irrational  when  all  the  other  terms  of  the  resulting 
r 

equation  were  rational.     No  such  term,  then,  could  occur.     In  the  same  way 

it  may  be  shown  by  operating  twice  on  (1)  with  V^  that  there  can  be  no  term 

»'^*Sv,_3  in  (1);  and  thus  step  by  step  we  can  reach  the  result  formulated  in  (1). 

3.  Express  x-yz  in  the  form  Si  +  r^S.2  +  ?'*'%• 

Suggestion:  tet  x^yz  —  Si-\-  r^S.  +  >'^So 

and  take  V^  of  both  members  we  get 

2yz  =  US,  +  20r^So. 
Operate  again  with  V^.  0  =  120>S'„.  Whence 

S,^(),     S,  =  \yz,     and     S,  =  )^{Qx-' -f  -  z')yz. 

4.  Express  sin^  B  cos-  0  sin  ^  cos  <^  in  terms  of  Surface  Spherical  Harmonics. 

Suggestion :  sin-  6  cos^  6  sin  <^  cos  <^  =  — ^  • 

Eor  result  v.  Art.  108  (3). 

111.  A  transformation  of  coordinates  to  a  new  set  of  axes  having  the  same 
origin  as  the  old  set  will  change  a  given  Surface  Spherical  Harmonic  into 
another  of  the  same  degree.  Eor  such  a  transformation  does  riot  change  the 
form  of  Laplace's  Equation  V^F=0  if  both  sets  of  axes  are  rectangular, 
and  it  is  effected  by  replacing  x,  y,  and  z  in  the  Solid  Harmonic  correspond- 
ing to  the  given  Surface  Harmonic  by  x  cos  a^  -\-  y  cos  a^  +  z  cos  a^, 
X  cos  (3i-{-y  cos  jSa  -f-  s:  cos  jSs ,  and  x  cos  yi  +  y  cos  yo  +  «  cos  yg  respectively, 
where  the  cosines  are  the  direction  cosines  of  the  new  axes,  and  it  will  leave 


Chap.  VI.]  LAPLACIANS.  209 

the  function  a  homogeneous  function  of  the  mth  degree  in  the  new  variables, 
and  on  dividing  this  by  the  vith  power  of  the  unchanged  radius  vector  we  shall 
have  a  Surface  Spherical  Harmonic  of  the  mth  degree. 

112.  We  have  seen  in  Art.  75  that  if  (x^,  y^,  «i)  are  the  coordinates  of  a 
given  point 

V  =  ,  ^  -^  (1) 

is  a. solution  of  Laplace's  Equation  V'^r^O,  and  transforming  to  spherical 
coordinates  that 

V=   .  -  ^  (2) 

V?-2  —  2?Vi[cos  6  cos  6 1  +  sin  B  sin  di  cos  (<^  —  <^i)]  +  *f 
is  a  solution  of 

rD:^{r  V)  +  ^^  A(sin  0  D^  V)  +  ^^  D^  V^  0  .  (3) 

If  y  is  the  angle  between  the  radii  vectores  /■  and  r^  of  the  points  (x,  y,  z) 
and  (j-i,  ?/i,  «i)  (1)  can  be  written 

V=^  ,  ^  (4) 

\)'-  —  2r)\  cos  y  +  i'l 

which  must  be  equivalent  to  (2),  and  hence 

cos  y  =  cos  6  cos  di  +  sin  6  sin  Ox  cos  (^  —  <^i)  . 

(4)  which  is  a  solution  of  (3)  is  of  the  same  form  as  (5)  Art.  75  and  by 
developing  it  as  we  developed  (5)  Art.  75  we  find  that 

r=P,„(cosy) 
is  a  solution  of  the  equation 

m(m  +  1)  F  +  ^  A(sin  d  A  V)  +  ^-^  D^  V=  0  (5) 

and  that  F=  r"'P,„(cos  y)     and      r=  ^,;^  P^(cos  y) 

are  solutions  of  (3). 

If  we  transform  our  coordinates  keeping  the  origin  unchanged  and  taking  as 
our  new  polar  axis  the  radius  vector  of  (xi,  y^-,  %).  y  becomes  our  new  Q  and 
P„j(cos  y)  reduces  to  P„j(cos  &) ,  a  Surface  Zonal  Harmonic,  or  a  Leyendrian*  of 
the  mth.  degree.  It  is  then  a  Legendrian  having  for  its  axis  not  the  original 
polar  axis  but  the  radius  vector  of  (x^,  y^,  z-^.  Since  a  Legendrian  is  a  Sur- 
face Spherical  Harmonic, 

P^(cos  y)  =  P,„[cos  6  cos  6i  +  sin  6  sin  6i  cos  (<^  —  <^i)J 
is  a  Surface  Spherical  Harmonic  of  the  mth  degree. 

«  V.  Art.  74. 


210  SPHERICAL    HAltMOXICS.  [Art.  113. 

It  is,  however,  of  yery  special  form,  since  being  a  determinate  function  of 
fi,  <f>,  /Ml,  and  (f>i  it  contains  but  two  arbitrary  constants  if  we  regard  it  as  a 
function  of  fx  and  ^,  instead  of  containing  27)i  +  1. 

It  is  known  as  a  Lajdace's  Coefficient,  or  briefly  as  a  Laplacian,  of  tlie  ?/ith. 
degree. 

We  shall  soon  express  it  in  the  regulation  form  of  a  Surface  Spherical 
Harmonic. 

The  radius  vector  of  (x^,  y^,  z^)  is  called  the  axis  of  the  Laplacian  and  the 
point  where  the  axis  cuts  the  surface  of  the  unit  sphere  is  the'^^oZem  _the 
Laplacian. 

We  shall  represent  the  Laplacian  P,„(cos  y)  by  L„^(/j,,  ^,  /^i.  <^i).  Of  course 
Ljjjx.  <f>,  1 ,  <^i)  ^  Pm(f^)  ^=  -Pm(cos  0),    and  is  really  independent  of  <^. 

113.     If  the  product  of  a  Surface  Spherical  Harmonic  of  the  mth  degree  by  a 

Laplacian  of  the  same  degree  is  integrated  over  the  surface  of  the  unit  sphere,  the 

47r 
result  is  equal  to  - — — —   multiplied  hy  the  value  of  the  Spjherical  Harmonic  at 

the  pjole  of  the  Laplacian. 
That  is, 

pz.^ Jr,,(^,  <^:)L^{iM,  ct>,  fMi,  ^,)d/ji  =  ^^i^  F„//xi,  <^o .  (1) 

Ti        -1 

Transform  to  the  axis  of  the  Laplacian  as  a  new  polar  axis,  and  let  -^^(At,  <^) 
be  the  transformed  Spherical  Harmonic.  L^^^/x,  ^,  fXi,  <^i)  will  become  P^(/a), 
and  (1)  will  be  proved  if  we  can  show  that 

f.li.fz„(t^,  «)P„W,V  =  ^^  ZJl,  0)  .  (2) 

0  —1 

.        Z„,{/,,  <i>)P,M  =  AlP„,{/x)J  +X^A,  cos  ncj.  +  B,  sin  n<^)P,;;(/x)P„.(/.) 
(V.  (5)  Art.  102). 

Jz,„ (/x,  <}>)Prn(f^)d<}>  =  27rAlF^(,x)Y  and 

0 


fd/.yz„Xf^,  cf>)PU,x)dct>  =  2^^  A         (V.  (5)  Art.  89) 


But  Z„,(l,  0)  =  Ao,  since  P„,(l)  =  1  and  P;;,(l)  contains  (1  —  1)1  as  a  factor 
and  is  equal  to  zero. 
Hence  (2)  is  proved. 


« 


Chap.  VI.]  LAPLACIANS.  211 

114.     We  can  now  express  a  Laplacian  in  the  regulation  form  as  a  Spherical 
Harmonic,  by  the  formulas  of  Art.  107. 

X,„(/A,  (j),  fii,  <^i)  =  P„,(cos  y)  =  P„,[cos  e  cos  0^  +  sin  8  sin  Oi  cos  (<^  —  ^i)J 


where 


2m  -\-  1  r*       /* 


= ^^  2;;rTi  ^-^^'^  =  ^-^^^^        ^^  ^^^  ^''-  ^^^' 


-^^^^  (m  +  ^oJ       J     ^^^'  '^'  ^"  '^'^  "^^^  n<pPZ(^)dfi 

2(m  -  70 !  ^^g  „^  p  v„  )  by  (1)  Art.  113,  and 


2??i  +  1   (7?i  —  ^^)  •'    ?•  7       /*  7-  v       . 

"•'"  ""      LV      (wi  4-  7i)  J    *^J     ™*^^'  *^'  ^1'  *^^^  ^"'  nxf>P:Xf^)dfx, 

2(ni  —  ??.) ! 


2m-ML  (mj— w)_! 

^4-^0!. 

^,„  +  ,,)  I  sin  n4,F,:(fx^)  by  (1)  Art.  113, 

and   J,„,t-  =  -'^H.A-  =  A,i-  =  0    by  Art.  105  unless  k^m .     Hence 

i,„(/x,  c/>, ;.!,  </.o = p,mp,,(/mo + 2X[|J^^;  P:n(f^)P:(H'i)  cos  7^(<^-  c^o]  •  (1) 

Each  term  of  a  Laplacian  involves  a  numerical  coefficient,  a  factor  which  is 
a  function  of  fi,  a  second  factor  which  is  the  same  function  of  /Xi ,  and  a  third 
factor  which  is  of  the  form  cos  k('^  —  <^i).  We  give  on  the  next  page  a  table 
of  the  first  few  Laplacians, taken  from  Minchin's  Statics,  omitting  in  each  term 
for  the  sake  of  brevity  the  function  of  /jl^  . 

By  the  aid  of  (1)  we  can  write  (5)  Art.   107  more  compactly.     It  becomes 

m  =  0  0  - 1 

m  =  »    2k  tt 


212 


SPHEPaCAL    HARMONICS. 


[Akt.  114. 


^ 

1 

-. 

:§; 

^k 

03 

1 

O 

o 

^ 

"o 

ccl- 

■-1^ 

o 

=^ 

. . 

■e- 

1 

")- 

~^ 

.3; 

"sr 

'\ 

Jc' 

=t 

1 

S 

1 

1 

o 

■ — • 

:t 

"o 

LCI  CO 

J?l- 

IB 

o 

o 

1 

1 

3 

ST 

'^i 

^' 

C<l 

=5. 

1 

1 

t^ 

^ 

8 

^ 

'=i. 

o 

CCiTt* 

21^ 

1 

'*^ 

~ — • 

o 

i.o|~ 

" 

,^ 

^ 

1 

1 

'l 

1     f 

It 

"s. 

o 

, 

1           J^ 

"« 

'3. 

^k 

o 

-  s 

1 

1 

"S 

•^ 

o 

Ct  |3C 

>--  1  X 

^^^ 

■©^ 

ct' 

1 

^^ 

+ 

s 

3 

"-^ 

CC 

s. 

o 

o" 

1 

1 

§ 

rH              a.               CJ 

g 

^ 

"s. 

1 

t<-. 

w 

"s. 

o 

—  I't' 

^ITJH 

i 

t*^ 

§ 

-IS 

-.              —                   N 

M 

»:!      k::        ►-; 

k;; 

4 

CilAP.  VI.]  SOLUTION    BY    DIRECT    INTEGRATION.  218 

EXAMPLE. 

Work  the  problems  of  Art.  108  and  Art.  108  Exs.  1  and  2  by  the  aid  of  (3) 
Art.  114. 

115.  Such  problems  as  we  have  handled  in  Arts.  98  and  99,  and  also  prob- 
lems differing  from  them  in  not  having  circular  symmetry  about  an  axis,  can 
now  be  solved  by  direct  integration. 

For  instance  let  it  be  required  to  find  the  value  at  an  external  point  of  the 
potential  function  due  to  the  attraction  of  a  solid  sphere  whose  density  at  any 
point  is  proportional  to  the  product  of  any  power  of  the  radius  vector  by  a 
Surface  Spherical  Harmonic. 

Let  p  =  Cv^  Y,n{l^ii  4>i)  • 

Then  using  our  ordinary  notation  we  have 


«  27r  1 

r      r      - 1  y>-'^  —  2r>-i  c( 


cos  y  +  ?f 

But  by  (3)  Art.  77 


1  1  r  r, 

,  =  -     Po(cos  y)  +  -  Pi  (cos  y) 

V/r'^  -  2rr,  cos  y  +  rf       r  L    °^        ^^  ^  >'      '^        ^' 

+  ^;  P,(cos  y)  +  •  ■  •  +  ^"  P,„(cos  y)  +  •  •  •] 

if    r>r,. 
Consequently  since 

iTt  1 

0  - 1 

V  reduces  to  the  single  term 

a  277  1 

(I  0-1 


47r(7         a"'  +  ^  +  3       Y„,(fi,cl>) 


2m  +  1    m-\-  k  +  3 


+ 1 


214  SPHERICAL    HARMONICS.  [Art.  115. 

EXAMPLES. 

1.  Solve  by  direct  integration  the  problems  worked  in  Arts.  98  and  99  and 
Examples  1,  2,  3,  and  4  of  Art.  99. 

2.  The  density  of  a  solid  sphere  is  proportional  to  the  product  of  the 
squares  of  the  distances  from  two  mutually  perpendicular  diametral  planes; 
find  the  value  of  the  potential  function  at  an  external  point. 

Ans.     p  =  A-/f  cos"^  6i  sin-  6i  cos"^  ^i 


kr 


\'  [^  ^o(y^i)  +  ^  A(i^O  +  ^  cos  2</„P|(/xO 

-^'(l^^(^)-ir9^^^-'^^^w)]- 

3.  Solve  Example  2  by  an  extension  of  the  method  of  Arts.  98  and  99. 

4.  A  conducting  sphere  of  radius  a  connected  with  the  ground  by  a  wire  is 
placed  in  the  field  of  force  due  to  an  electrified  point  at  which  in  units  of 
electricity  are  concentrated.  Find  the  value  of  the  potential  function  due  to 
the  induced  charge. 

Suggestion:  Let  Vi  be  the  potential  function  due  to  the  point,  and  V.^  that 
due  to  the  induced  charge,  and  let  b  be  the  distance  of  the  point  from  the 
centre  of  the  sphere.     Then 


^b-'  —  2br  cos  6  +  ?-2 
=  j^  [PoCcos  6)  +  '-  Pi  (cos  0)  +  p  Po(cos  ^)  +  •  •  -I     if     r<b. 

=  ^'[Po(cos^)+^Po(cos^)+^Jp,(cos^)  +  ---]     if     ^■>^>- 
Vo  =  AoPo(cos  ^)  +  J 1  -  Pi  (cos  0)  +  ^2  -  Po(cos  0)  +  ---     if     r<a. 

=  Ao-  Po(cos  e)  +  Ji  ^  Pi  (cos  6)+Ao_%  P2(cos  ^)  +  •  •  •    if    r>  a. 

When    r  =  a     V^  +  1^  =  0  .     Hence 

m  ma  ma^ 

Ao  —  —  —,     A^———,     A2—         TT'"* 


Chap.  VI.]  AXES    OF    A    SPHERICAL    HARMONIC.  215 

and 

F2=-^rPo(cos^)+^Pi(cos^)  +  pP2(cos^)  +  ---]     ii'     r<a 

=  -^rPo(cos^)  +  ^'Pi(cos^)+|^,Po(cos^)  +  --'1     if     T>a. 

Hence  the  effect  of  the  induced  charge  is  precisely  the  same  at  an  external 
point  as  if  the  sphere  were  replaced  by  -^  units  of  negative  electricity  con- 
centrated  at  the  point  /•  =  — ,  ^  =  0 .     v.  Peirce,  Newt.  Pot.  Punc,  §  66. 

116.  If  the  two  points  P  and  P'  are  taken  on  the  line  OH  whose  direction 
cosines  are  \,  fx,  and  v,  and  if  u  and  u'  are  the  values  at  P  and  P'  of  any  con- 
tinuous function  of  the  space  coordinates,  then  .  is  called  the 

partial  derivative  of  ti  along  the  line  OH  and  will  be  represented  by  D/,u. 

Let  X,  y,  z  be  the  coordinates  of  P  and  x  +  \x,  y  -\-  Ay,  z-\-  ^z  the  coordinates 
of  P';  then 

w'  —  u  =  D^u.i\x  +  DyU.Ay  +  Z>,  u.Az  -f-  e 

where  e  is  an  infinitesimal  of  higher  order  than  the  first  if  Ax,  Ay,  and  As  are 
infinitesimal  (v.  Dif.  Cal.  Art.  198). 

2i'  —  u        _  A.i-     ,     ^^         A?/     ,    ^         Az     ,       e 

Hence         -^^  =  P*^  u.  ^^  +  D^  u.  ^^  +  P,  u.  ^^  +  -^^  • 

T>   ,  Acc        ^        Ay  T      A« 

But  :=  A ,    — ^  =  u ,    and    =  v . 

PP'       ^'    pp'       -"'    ''"''   PP' 

Therefore  D,^  u  =  XD^  n  +  /xP^  «  +  vP, « .  (1) 

If  V^i/^0,  D'^D^D^u  is  a  solution  of  Laplace's  Equation. 
Por  \7'XPl'P>^D;uj  =  P^'P;P;(  V  ?0  =  0  • 

Hence  if  V^w  =  0  P^?<  is  a  solution  of  Laplace's  Equation,  and  if  OHi, 

OH2,  OHs,  •  •  •  are  a  set  of  lines  through  the  origin  P^^P^^P^^-  --u  is  a  solution 

of  Laplace's  Equation. 

117.  If  H/^.  is  a  rational  integral  homogeneous  Algebraic  function  of  x,  y, 
and  z  of  the  A;th  degree 

IxH,       g,_,^      IxH,      ,-'g,._, 

yi+2      r"        ^.i  j.i  +  2    "T      ,^.i  +  3      » 

rr 

and  is  of  the  form  -^^ . 


216  SPHERICAL    HARMONICS.  [Art.  118. 

The  same  thing  can  be  proved  of  Dy  (-7  )  and  D^  \~t)  and  therefore  liolds 
good  of  A  (f'). 

If  w  is  a  homogeneous  function  of  x,  y,  and  z  of  the  degree  —  m  —  1   and 
V2?<  =  0  then  V^(>-'"  +  ^w)=0. 

\j2^^sm  + 1  „)  ^  ^2m  +  1) (2m  +  2)/-2'»- ^  u 

+  2(2m+  l)r2'»-i(a;D^?t  +  //D^^w  +  zD^u)  +  /•2'»  +  iV^it 
=  0, 
since  xDj.u  +  ?/-0j^?^  +  si)^?/  =  —  (m  +  l)?^ 

by  Euler's  Theorem  (v.  Dif.  Cal.  Art.  220). 

118.     —  =  =^=i     is  a  solution  of  Laplace's  Equation  (v.  Art.  75) 

'•        S/x'  +  t/  +  z' 

TT 

and  is  of  the  form  — -  • 
r 

an 


D^^JD^^D^^'--D^^i^)    is  then  a  solution  of  Laplace's  Equation  by  Art.  116; 

is  of  the  form 
degree  —  in  —  1 , 


TT 

it  is  of  the  form    ^J'^  ^  by   Art.    117   and   is   a  homogeneous   function   of  the 


Therefor 


■e     ?" 


^I),,^D/^^B^^---D^^l—)     is  a  solution  of  Laplace's  Equation, 

and  is  a  rational  integral  homogeneous  Algebraic  function  of  x,  y,  and  z  of  the 
mth    degree,  and  is   consequently  a   Solid   Spherical    Harmonic  of   the    ???th 

degree  (v.  Art.  110) ;  and  7-"'  +  ^AiA.Ag- ' "  A,„  (^)    is  a  Surface  Spherical  Har- 
monic of  the  ??ith  degree. 

Moreover  since  the  direction  of  each  of  the  lines  OHi,  OH^,  •  ■  •  0H„^  depends 
upon  two  angles  which  may  be  taken  at*pleasure,  these  angles  and  M  are 

27n  +  1  arbitrary  constants  and  may  be  so  chosen  that     r"'-^^D^  D^_^-  •  ■  I)^^^  I  —  j 
may  be  any  given  Surface  Spherical  Harmonic. 

Consequently  any  given  Surface  Spherical  Harmonic  may  be  regarded  as 

formed  by  differentiating  ^  successively  along  m  determinate  lines  OH^,  OH^  •  •  • 

OH^,  and  is  given  except  for  the  undetermined  factor  ilf  when  these  lines  are 
given. 

The  lines  OH^,  OH^,  OH^,  •  •  •  Olfj^  are  called  the  axes  of  the  Harmonic,  and 
the  points  where  they  meet  the  surface  of  the  unit  sphere  the  j^oles  of  the 
Harmonic.  The  m  axes  of  a  Zonal  Harmonic  coincide  with  the  axis  of  coordi- 
nates (v.  Art.  86)  and  consequently  the  m  axes  of  a  Laplacian  coincide  with 
what  we  have  called  the  axis  of  the  Laplacian  (v.  Art.  112). 


CiiAP.  VI.]  ROOTS    OF    ZONAL   AND   TESSERAL   HARMONICS.  217 

119.  Any  Surface  Zonal  Harmonic  PrtiitJ)  is  equal  to  zero  for  m  real  and 
distinct  values  of  /x  which  lie  between  —  1  and  1 ;  and  any  Associated  Func- 
tion Pliii)  is  equal  to  zero  for  m  —  n  real  and  distinct  values  of  y.  which  lie 
between  —  1  and  1 . 

—^ — : — —  contains   (fx-—  Xy-''  as  a  factor,     v.  Art.  89. 

From  Eolle's  Theorem,  "  If  f{x)  is  continuous  and  single-valued  and  is  equal 

df(x) 
to  zero  for  the  real  values  a  and  h  of  x,  -"^-^  is  equal  to  zero  for  at  least  one 

real  value  of  x  between  a  and  i,"  (v.  Dif.  Cal.  Art.  126)  it  follows  that  since 

d(a-  —  1)'" 
(/x-  — 1)"'  =  0   when    /^  =  —  1    and  when    /x  =  1      ^ =0    for  at  least 

one  value  of  u  between  —  1  and  1.     -^'^—-, —    cannot    be    equal  to  zero  for 

'^  dy 

more  than  one  value  of  \x  between  —  1  and  1,  for  it  contains  {y^  —  1)'"~^  as  a 

factor  and  is  a  rational  Algebraic  polynomial  of  the  2m  —  1st  degree. 

In  like  manner  we  can  show  that    ^    ,  ^ — —  =  0  has  m  —  2  roots  equal  to 

d/x'' 

—  1,  111  —  2  roots  equal  to  1  and  two  real  roots  between  —1  and  1  which 
separate  the  three  distinct  roots  of    ^      —  =0;  and  in  general  if  k  <  ?/^  + 1 

that  — ^^-, ^0  has  m  —  k  roots  equal  to— 1,   m  —  k  roots  equal  to  1, 

'^-^                                                                               rf^-Yu--l)"' 
and  k  real  roots  separating  the  k  +  1  distinct  roots  of  jlT^i ~  ^  • 

\  clm(fj;i  —  Y)in  ^^ 

Hence  P„Yu)  =  0  or ^   , —  =0  has  in  real  and  distinct  roots 

'"^^  2'"»i!  rt/u,'" 

between  —  L  and  1,  and  it  has  no  more  since  it  is  of  the  int\\  degree. 

The  same  reasoning  shows  that ; — =  0    has  m  —  n  distinct  real 

dfx'"-^" 

roots  between  —  1  and  1,  and  therefore  that  P,'',(/U,)  is  equal  to  zero  for  m  —  )i 
distinct  real  values  of  /x  between  —  1  and  1.  Since  P,;;(/u)  contains  sin"  6  as  a 
factor  it  is  also  equal  to  zero  when  /x  =  —  1  and  Avhen  fi^l . 

cos  n<f>  is  equal  to  zero  for  2?i  equidistant  values  of  <^,  and  sin  n<f>  is  equal  to 
zero  for  2>i  values  of  <^.  Hence  any  Tesseral  Harmojiic  sin  n<f>P;,',(iJi)  or 
cos  Ncl>P^[(fx)  is  equal  to  zero  for  2n  equidistant  values  of  <^,  for  /x  ^=  1,  for 
/x  =  —  1,  and  for  i7i  —  n  real  and  different  values  of  /x  between  —  1  and  1. 

It  follows  that  the  value  of  any  Surface  Zonal  Harmonic  -P,„(/x)  at  a  point 
on  the  surface  of  the  unit  sphere  will  have  the  same  sign  so  long  as  the  point 
remains  on  one  of  the  zones  into  which  the  surface  of  the  sphere  is  divided  by 


218  SPHERICAL    HARMONICS. 

the  m  circles  of  latitude  corresponding  to  the  m  roots  of  Pm(H')  =  ^?  ^^^  "^^^^^ 
change  sign  whenever  the  point  passes  from  one  of  these  zones  into  an  adjoin- 
ing one;  and  that  the  value  of  any  Tesseral  Harmonic  sin  7i(f>P,l(fM)  at  a  point 
on  the  surface  of  the  unit  sphere  will  have  the  same  sign  so  long  as  the  point 
remains  on  any  one  of  the  tesserae  into  which  the  surface  of  the  sphere  is 
divided  by  the  m  —  n  circles  of  latitude  corresponding  to  the  roots  of  P/,|(//.)  =  0 
and  the  2n  meridians  corresponding  to  the  roots  of  sin  n<f)  =  0,  and  will  change 
sign  whenever  the  point  passes  from  one  of  these  tesserae  into  an  adjoining 
one. 


CHAPTER   VII.* 

CYLINDRICAL    HARMONICS    (BESSEL's    FUNCTIONS). 

120.     In  Arts.  11  and  17  we  obtained 

z  =  AJ,(x)  +  BKo(x)  (1) 

as  the  general  solution  of  Fouriei-^s  Equation 

where  j-„(a.)  =  l_|^  +  ^^__£_+...  (3) 

and  is  called  ?i,  ^Cylindrical  Harmonic  or  BesseVs  Function  of  the  zeroth  order; 
and  where 

-^0(^0  =  ^0(^0  log  ^  +  I  -  2J4"2  (1  +  2)  +  2^1ij2  (1  +  2  +  3) ^^^ 

and  is  called  a  Cylindrical  Harmonic  or  BesseVs  Functioii  of  the  Second  Kind, 
and  of  the  zeroth  order. 

In  Art.  17  we  found  that  z  =  Jn(x) 

is  a  particular  solution  of  BesseVs  Fquation 

where  if  n  is  unrestricted  in  value 

^»(^")  =  2'T(/i+T)  L^  ~  2->  +  l)  +  2''.2!(/i  +  l)(7i  +  2) 

~  2«.3!(w  +  l)(«  +  2)(7i  +  3)  ""         J      ^^^ 

and  is  called  a  Cylindrical  Harmonic  or  BesseVs  Functio7i  of  the  wth  order ; 
and  that  unless  n  is  an  integer 

z  =  AJ„(x)  +  BJ_„(x) 

is  the  general  solution  of  Bessel's  E(]uation. 

*The  student  should   re-read  carefully  Arts.  11,  17,  and  18{cl)  before  beginning  this 
chapter. 


220  CYLINDRICAL    HARMONICS.  [Art.  121. 

If  n  is  an  integer  it  can  be  shown  that 

J-„(aO  =  (-l)"JL„(x), 
(v.  Forsyth's  Diff.  Eq.  Art.  102),  and  then 

is  the  general  solution  of  Bessel's  Equation  and 

{A-„(.)}  =  U.)  log  .  - 1  (g'^k^JL^  @^'- 

V.  M.  Bocher,  Ann.  Math.  Vol.  VI,  No.  4. 

121.  A  useful  expression  for  Jj,{x)  as  a  definite  integral  can  be  obtained 
without  difficulty  from  Bessel's  Equation  [(5)  Art.  120]  by  a  slight  modifica- 
tion of  the  method  given  by  Forsyth  (Diff.  Eq.  Art.  136). 

It  was  shown  in  Art.  17  that  z  =  x"v  is  a  solution  of  Bessel's  Equation  if 
V  satisfies  the  equation 

dx^  X       dx 


Assume 


=  CTQQ^{xt)dt  (2) 


where  x  and  t  are  independent,  T  is  an  unknown  function  of  t,  and  a  and  h 
are  at  present  undetermined. 

Then  ^  =  -  p T  sin  {xt)dt 

h 

d-v  r 

and  -^.,  =  -J  t-T  cos  {x£)dt . 

Substituting  in  (1)  after  multiplying  through  by  x,  we  have 
h  h 

("(1  —  f)Tx  cos  {xi)dt  —  ({'In  +  \)tT sin  {xt)dt  =  0 .  (3) 


Chap.  YII.]        BESSEL'S    FUNCTIONS    AS    DEFINITE   INTEGRALS.  221 

By  integration  by  jparts  we  find  that 

h  6 

((\  —  f)Tx  cos  {xt)dt  =  r(l  —  f-)T?iin  (xt)~\ 

a  (, 

^     ~/[^^  ~  ^'^  Tt~  ^^^]  '"'  ^^^'^^^^ ' 

and  (3)  reduces  to 

r  (1  -  f)  T  sin  {xt)  1  - j"|^  (1  -  t'i^  +  (2m  -  l)i^r1  sin  {xt)dt  =  0.       (4) 

If  we  determine  T  so  that 

a-f')^+(^>^-'^)fT=o,  (5) 

and  a  and  ^»  so  that  (1  —  t^)T  sin  (xt)\  =  0  (6) 

(4)  will  be  satisfied  and  our  problem  will  be  solved.     (5)  gives 

T=C(l-t^r-i,  (7) 

and  (6)  will  obviously  be  satisfied  if  re  =  —  1  and  b  =  l. 

^,  r(l  —  f^Y  cos  (xt)df  .  ,     .         ,    ,, 

Hence  v=  C  \  ~ ; —      ^     —  is  a  solution  of  (1), 

I 
and  z=C X"  r(l  -  tr  cos  (xt)dt  3. 

is  a  solution  of  Bessel's  Equation. 
If  we  let  t  =^  cos  ^  in  (8)  we  get 


C  x"  I  sin'^"  fji  cos  (x  cos  cf>)d<f>. 


Expand  cos  (x  cos  ^)  into  a  series  involving    powers  of  x  cos  <^,  integrate 
term  by  term  by  the  aid  of  the  formulas 

f  sin"  x.dx  =  ^-    ^    -    ,        [Int.  Cal.  (1)  Art.  99], 

^  '    rg  +  i) 


222  CYLINDRICAL   HARMONICS.  [Art.  122. 


I  sin"  X 


cos'"  x.dx  = 


2r(^'+i) 

(Int.  Cal.  Art.  99  Ex.  2),  and  compare  with  (6)  Art.  120,  and  we  get 

J^(x)  = —  I  sin^"  <fi  cos  (x  cos  <^)(/c/).  (9) 

2"V^r(»  +  2y 

If  ?i  is  a  positive  integer  (9)  reduces  to 
1  "  ^ 

Let    n  =  0   in  (9)  or  (10)  and  we  get 

Jo(^)  =  —   I  cos  (X  cos  (^)fZ^.  (11) 

EXAMPLES. 

1.  Obtain  Formula  (11)  directly  from  Fourier's  Equation,  (2)  Art.  120. 

2.  Prove  by  integration  by  parts  that  if  ?i  >  —  - 

I  sin-"  <^  cos  (^  sin  (x  cos  <^)f/c^  =  ^ — -j^  j  sin-"  +  -  ^  cos  (ic  cos  (f>)d(j>. 

0  II 

3.  Prove  by  integration  by  jmrts  that  if  n"^  - 
I  sin-"  ^  cos  <f)  sin  (.:<:•  cos  <^)''''<^ 

0 

1  r 

=  -i[2n  sin-"  <^  —  (2n  —  1)  sin^"  -  2  <^]  cos  (x  cos  <^)f/<^ . 

u 

122.     We  can  now  readily  obtain  a  number  of  useful  formulas. 
Differentiate  (11)  Art.  121  with  respect  to  x  and  we  get 


dJo(x) 
dx 


—  ^ I  cos  ^  sin  (x  cos  ^)fZ<^ 

II 

= I  sin-  ^  cos  (x  cos  <^)6?0        by  Ex.  2  Art.  121. 


Chap.  VII.]  PKOPEKTIES    OF    BESSEL's    FUNCTIONS.     .  ±1'^ 

Hence  by  (10)  Art.  121  ^^^  =  -  J,{x).  (1) 


dx 


In  like  manner  by  the  aid  of  Exs.  3  and  2,  Art.  121,  we  can  obtain  the 
relations 

d[x"JJx)~\ 


.  dx 

if  n>      , 


if  n>  ~\ 


^C^=^  =  —'...(.)  (3) 


(2)  can  be  written 

Jx"J,,  _ ,  (x)dx  =  x"J,  (x)  (4) 


if  71  >  ~ 


(2)  and  (3)  can  be  written 

and  X-  "  ^^  -  nx-  "  -  './„  (.r)  =  -  .7-  "J,,  ^  ^  (x), 

^-^-.(^)-^^(^)  (5) 


whence  2  ?^  =  ,/„  _,(..)  -  ,/„  ^  .(.r) 


and 


(J) 


Jn{^)=jn-.{X)+'1.^,{X).  (8) 


The  repeated  use  of  formula  (8)  will  enable  us  to  get  from  J^ix)  and  Ji(x) 
any  of  Bessel's  Functions  whose  order  is  a  positive  integer.  For  example,  we 
have 

-^3(-r)  =  (|-l)^i(^-)-^^o(^). 


22-i  CYLIXDillCAL    HAKMONICS.  [Art.  122. 

From  a  table  giving  the  values  of  Jo(x)  and  Ji(x),  then,  tables  lor  the 
functions  of  higher  order  are  readily  constructed.  Such  a  table  taken  from 
Eayleigh's  Sound  (Vol.  I.,  page  265)  will  be  found  in  the  Appendix  (Table  VI.). 

By  the  aid  of  (5)  and  (6)  any  derivative  of  Jn(x)  can  be  expressed  in  terms 
of  J„(x)  and  J„  +  i(x).     For  example 

If  we  write  Jo(x)  for  z  in  Fourier's  Equation  [(2)  Art.  120],  then  multiply 
through  by  xdx  and  integrate  from  zero  to  x,  simplifying  the  resulting  equa- 
tion by  integration  by  payts,  we  get 

x'^^^^  ^xJ,(x)dx  =  ^; 
whence  by  (1)  ( xj,,{xyx  =  xJ^(x) .  (9) 

If  we  write  Jq(x)  for  z  in  Fourier's  Equation,  then  multiply  through  by 
^2  — ^LJ  ^x  and  integrate  from  zero  to  x,  simplifying  by  integration  by  parts 
we  get 

whence  by  (1)         ^xi,T,{x)Ydx  =  |'  [(Jo(^-))'^  +  ('A(-^))^].  (10) 

In  like  manner  we  can  get  from  Bessel's  Equation  [(o)  Art.  120]  the  formula 

Jx(J^(x)ydx  =  \  [.r^  (^^)  +  {^'  -  nWn{^)y~]  (11) 

0 

which  (6)  enables  us  to  reduce  to  the  form 

J;r07„(a;))2fZ^  =  |'[(J-„(:r))^+  (.7„^i(x))-^]  -..a-J„(^K^i(a;).  (12) 

Formulas  (9),  (10),  (11),  and  (12)  will  prove  useful  when  we  attempt  to 
develop  in  terms  of  Cylindrical  Harmonics. 


Chap.   VII.J  PROPERTIES    OF    BESSELS    FUNCTIONS.  225 

Values   of  J„(oc)   for   Larger   values   of  x  than   those   given   in   Table  III., 
Appendix,  may  be  computed  very  easily  from  the  formula 


2  !  (8.')--' 


_  +^ — 4178^0^ — rx-i-"v 

3!73^)-3^ +  ■  ■  •  J"K"  -4  -  ''V-         (1'^  > 

v.  Lommel,  Studien  fiber  die  Bessel'schen  Functionen,  page  59. 

The  series  terminates  if  2n  is  an  odd  integer,  but  otherwise  it  is  divergent. 
It  can  be  proved,  however,  that  in  any  case  the  sum  of  7n  terms  differs  from  J„(x) 
by  less  than  the  last  term  included,  and  consequently  the  formula  can  safely 
be  used  for  numerical  computation. 

EXAMPLES. 

1.  Confirm  (1),  (2),  and  (3),  Art.  122,  by  obtaining  them  from  (3)  and  (6), 
Art.  120. 

2.  Confirm  (1),  Art.  122,  by  showing  that  Fourier's  Equation  will  differ- 
entiate into  the  special  form  assumed  by  Bessel's  Equation  when  w  =  1. 

3.  Show  that  (9),  Art.  122,  is  a  special  case  of  (4),  Art.  122. 

4.  Show  that  the  limit  approached  by  J„(x)  as  n  increases  indefinitely  is 
zero,  and  by  the  aid  of  this  fact  and  of  (8),  Art.  122,  prove  that 


^-i(-^-)  =;[^^^«(-^-)  -  (:n  +  2)J„^,(x)  +  (,^  +  A)J,^,(x)  +  •••], 


2 

X 

5.    Prove  that 

dJ„(x)       2 


dx  .ii^^^"('^">  -  ('^  +  2).^„,,(a')  +  (M  +  4).7-„,,(x-)  -•..]. 

6.    Show  that  the  substitution  of  ( 1  —  —,)    for  x  in  Legendre's  Equation 
will  reduce  it  to  the  form 

\  //-/  (h/-        \//       n-f  an       \         nf 

and  that  the  limiting  form  approached  by  this  equation  as  n  is  indefinitely 
increased  is  Fourier's  Equation,  and  hence  that  Jq{x)  can  be  regarded  as  some 

constant  factor  multiplied  by  the  limiting  value  approached  by  -P„(l— — j" 
as  n  is  indefinitely  increased. 


226  CYLINDRICAL    HARMONICS.  '         [Art.  123. 

To   complete    the   solution   of   the   drumhead    problem  taken   up   in 
Art.  fl,  ,\"  found  that  it  would  be  necessary  to  develop  a  given  function  of  r 

in  the  form 

/'(/•)  =  Ayl,(fx,r)  +  .LJ,(fi,r)  +  A,J,{fi,r)  +  •  •  • 

where  /Xj,  /Ug,  /A3,  &c.,  are  the  roots  of  the  transcendental  equation  jQ^^fxa)  =  0  ; 
and  in  Art.  11,  Ex.  the  development  of  unity  in  a  series  of  precisely  the 
same  form  was  needed. 

(o)    Let  us  consider  another  problem. 

The  convex  surface  and  one  base  of  a  cylinder  of  radius  a  and  length  h  are 
kept  at  the  constant  temperature  zero,  the  temperature  at  each  point  of  the 
other  base  is  a  given  function  of  the  distance  of  the  point  from  the  centre  of 
the  base  ;  required  the  temperature  of  any  point  of  the  cylinder  after  the 
permanent  temperatures  have  been  established. 

Here  we  have  to  solve  Laplace's  Eqiuation  in  Cylindrical  Coordinates 
([xiv]  Art.  1). 

ir-u  +  \  J>,.n  +  ]:,  l>ln  +  l:n  =  0  (1) 

subject  to  the  conditions 

?(  =  0      when  ,-;  =  0 
K  =  0  ''       /•  =  a 

»=/(;•)     -       z  =  h, 

and  from  the  symmetry  of  the  problem  we  know  that  D^u  =  0. 
Assuming  as  usual  u  =  R.Z  we  break  (1)  up  into  the  equations 

■^  +  7-^  +  ^'^'='^' 

Avhence  u  =  sinh  (^fMz)J^(/jt.r)  (2) 

and  11  =  cosh  (/xz)J^(/xr)  (3) 

are  particular  solutions  of  (1). 

If  fi^.  is  a  root  of  f^oif^a)  =  0  (4) 

n.  =  sinh  (fx,^.z)J,,(fj.,,i-) 
satisfies  (1)  and  two  of  the  three  equations  of  condition. 

If  then  /(;■)  =  JA>(/^,r)  +  A„J,(fi,r)  +  A,J,(tM,r)  +  •  •  •  (5) 

fi^,  fXo,  /Xg,  &c.,  being  roots  of  (4), 

sinh  (/ii.~)  X    I     r   sinh  (fj...t)  sinh  (^3,-:) 

satisfies  (1)  and  all  of  the  equations  of  condition,  and  is  the  required  solution. 


Chap.  VIL]  FLOW    OF    HEAT    IN    A    CVLINDEK.  ^27 

(I))  If  instead  of  keeping  the  convex  surface  of  the  cylinder  at  thr  tempera- 
ture zero  we  surround  it  by  a  jacket  impervious  to  heat,  the  ("quation  of 
condition  »  =  0  when  r  =  «.  will  be  replaced  by  D,.u  =  0  when  r  =  a,  or  if 

u  =  sinh  (fxz)Jo(fir), 

by  — -ALL^  —  0  when     r  =  a, 

that  is  by  ^^^,'(^,,)  =  0  *  or  (v.  (1)  Art.  122) 

by  ^i(/ur/)=0.  (7) 

If  now  in  (5)  and  (6)  /Wj,  [x^,  /j.^,  &c.,  are  roots  of  (7),  (6)  will  be  the  solu- 
tion of  our  new  problem. 

(c)  If  instead  of  keeping  the  convex  surface  of  the  cylinder  at  the  tempera- 
ture zero  we  allow  it  to  cool  in  air  at  the  temperature  zero,  the  condition  u  =  0 
when  r  =  a  will  be  replaced  by  D,.n  -\-  ////  =  0  when  r  =  <(■,  or  if 

n  =  siuli  (/>i,?)t/y(/xr) 
by  /xJo(/xr)  +  /^/o(/x?-)  =  0  when     r  =  a 

"that  is  by  fiaj^'dxa)  +  ahJo(fia)  =0  or  (v.  (1)  Art.  122) 

by  fi((J^(fxa)  —  ahJ^t(fx,a)  =  0.  (8) 

If  now  in  (5)  and  (6)  /x^,  /Uo,  fis,  &c.,  are  roots  of  (8),  (6)  will  be  the 
solution  of  our  present  problem. 

124.     It  can  be  shown  that  Jo(x)  =  0  (1) 

M.r)=0  (2) 

and  a:Jo'(x)  +  A'/o(.r)  =  0  (3) 

have  each  an  infinite  number  of  real  positive  roots  (v.  Riemann,  Par.  Dif.  GL, 
§  97).  The  earlier  roots  of  these  equations  can  be  computed  without  serious 
difficulty  from  the  table  for  the  values  of  Jo(x)  (Table  VI.,  Appendix). 

The  first  twelve  roots  of  Jo(x)  ==0  and  Ji(x)  =0  are  given  in  Table  IV., 
Appendix,  a  table  due  to  Stokes.  Large  roots  of  Jo(x)  =  0  and  of  Ji(x)  =  0 
may  be  very  easily  computed  from  the  formulas 

^  _  .  _  9r   .  -050661  _  .053041        .262051  _ 

x^_      ,    ^^      .151982   ,    .015399        .245835    , 
^-'■^-'       4.S+1    +(4^  +  1)^      (4.  +  l)^^""  ^^ 

given  by  Stokes  in  Camb.  Phil.  Trans.,  Vol.  IX.,  x'-'^  representing  the  sth  root 
of  Jo(x)  =  0,  and  a-<f  the  .sth  root  of  J^(.r)  =  0. 

*  We  shall  find  it  convenient  to  use  the  familiar  notation  of /'(x)  =  -^-j—'  (v.  Dif.  Cal.,  p.  119). 


228  CYLINDRICAL    HARMONICS.  [Art.  126. 

125.     We  have  seen  in  Art.  123  that 

U  =  sinh  (fi^.z)jQ(^lj,/.r)  and  r=  sinh  (yLi,-:>/o(/^,/-)  are  solutions  of  V^U=0 
and  \7'^V=0  if  we  express  Laplace's  Equation  in  terms  of  Cylindrical 
Coordinates  (v.  (1)  Art.  123). 

Hence,  if  fdS  represents  the  surface  integral  over  any  closed  surface,  we 
have 

J(iW,V-  VD,U)dS^O 

by  Green's  Theorem  (v.  Art.  92). 

If  we  take  the  cylinder  of  Art.  123  as  our  surface,  and  perform  the 
integrations  and  simplify  the  resulting  equation,  we  find 

CrJ^,(fl^r)Jf^(^x^7•)d^'=— :;  [/i^o,  Jo(M;«)^/(^/;'0  "  A^/''-^o(/iA«)^)'(/^;«)] 

0 


Hence  if  jx^.  and  /Lt,  are  different  roots  of 

j;,(^a)  =  (), 

or  of  ^i(/^''0=*'; 

or  of  iJiaJy(fia)  —  \J^,{/j.'i)  =  0, 


then 


JrJ,(lx,r)J,(/xp-)dr  =  0.  (2) 


EXAMPLE. 

Obtain  (1)  Art.  125  directly  from  Fourier's  Equation 

di^^  r      dr  ^ 

126.     We  are  now  able  to  obtain  the  developments  called  for  in  Art.  123. 

Let  /(/•)  =  A,J,{^l,r)  +  A,J,{fi,r)  +  A,J,{^l,r)  +  •  •  •  (1) 

/*!,  /X2,  /As,  &c.,  being  roots  of    J(t(/xa)  =0,  or  of  Ji(fin)  =0,  or  of 

/xuJi(/jLa)  —  \Jf)(/jLa)  =  0. 

To  determine  any  coefficient  A^  multiply  (1)  by  r  Jf^{^fx^r)dr  and  integrate 
from  zero  to  a.     The  first  member  will  become 

jrf{r)J,{fi,r)dr. 


Chap.  VII.]  CYLINDRICAL   HARMONIC    SERIES.  229 

Every  term  of  the  second  member  will  vanish  by  (2)  Art.  125  except  the 
term 

A,fr(J,(fi,r)ydr. 

fr(J,(fi,r)ydr  =  -    fx(J,(x)ydx  =  '^  [Wf^,a)y  +  (^i(/^,«))'] 
by  (10)  Art.  122. 

«^"-      -^'- =,.t(^„(...))' +(■/,(.,>■))-]  fr'-^'^'^'-'-'^"-  (^> 

The  development  (1)  holds  good  from  r  =  0  to  r  =  a  (v.  Arts.  24,  25,  and  88). 
If  fii,  fio,  fi^,  &c.,  are  roots  of  Jq(/j-<')  =0,  (2)  reduces  to 

If  fii,  /j,2,  fji's,  &c.,  are  roots  of  Jii^fid)  =0,  (2)  reduces  to 

If  /Lti,  /i2,  yu-3,  &c.,  are  roots  of  fiaJi^/Mu)  —  XJ,^(/xa)  =0,   (2)  reduces  to 


For  the  important  case  where  /(;■)  =  1 

^^  0  "^    '  ^^ 

by  (9)  Art.  122,  and  (3)  reduces  to 

9 

J  — - ,  (7) 

(4)  reduces  to  J^.  =  0  except  for  k  =  l  when  i^f.  =  0  and  we  have  Ai=  ], 

2X 

(5)  reduces  to  -4„  =  ^^^,-j_-^^^^_^ .  (8) 


230  CYLINDRICAL    HARMONICS.  [Art.  127. 

EXAMPLES. 

1.  Show  that  in  (12)  Art.  11  any  coefficient  A^.  has  the  value  given  in  (3) 
Art.  126  ;  and  in  the  answer  to  Art.  11,  Ex.  the  value  given  in  (7)  Art.  126. 

2.  Show  that  if  a  drumhead  be  initially  distorted  so  that  it  has  circular 
symmetry,  it  will  not  in  general  give  a  musical  note  ;  that  it  may  be  initially 
distorted  so  as  to  give  a  musical  note  ;  that  in  this  case  the  vibration  will  be 
a  steady  vibration  ;  that  the  periods  of  the  various  musical  notes  that  can  be 
given  when  the  distortion  has  circular  symmetry  are  proportional  to  the  roots 
of  Jo(a')  =  0  ;  that  the  possible  nodes  for  such  vibrations  are  concentric  circles 
whose  radii  are  proportional  to  the  roots  of  t7o(.r)  =  0. 

3.  A  cylinder  of  radius  one  meter  and  altitude  one  meter  has  its  upper 
surface  kept  at  the  temperature  100°,  and  its  base  and  convex  surface  at  the 
temperature  15°,  until  the  stationary  temperature  is  set  up.  Find  the  tempera- 
ture at  points  on  the  axis  25  cm.,  50  cm.,  and  75  cm.  from  the  base,  and  also 
at  a  point  25  cm.  from  the  base  and  50  cm.  from  the  axis. 

Ans.,  29°.6  ;  47°.6  ;  71°.2  ;  25°.8. 

4.  An  iron  cylinder  one  meter  long  and  twenty  centimeters  in  diameter  has  its 
convex  surface  covered  with  a  so-called  non-conducting  cement  one  centimeter 
thick.  One  end  and  the  convex  surface  of  the  cylinder  thus  coated  are  kept  at  the 
temperature  zero,  the  other  end  at  the  temperature  of  100°.  Find  to  the  nearest 
tenth  of  a  degree  the  temperature  of  the  middle  point  of  the  axis,  and  of  the 
points  of  the  axis  twenty  centimeters  from  each  end  after  the  temperatures 
have  ceased  to  change.  Given  that  the  conductivity  of  iron  is  0.185  and  of 
cement  0.000162  in  C.  G.  S.  units.  Find  also  the  temperature  of  a  point  on 
the  surface  midway  between  the  ends,  and  of  points  on  the  surface  twenty 
centimeters  from  each  end.  Find  the  temperatures  of  the  three  points  "of  the 
axis,  supposing  the  coating  a  perfect  non-conductor,  and  again,  supposing  the 
coating  absent.     Neglect  the  curvature  of  the  coating. 

Ans.,  15°.4  ;  40°.85  ;  72°.8  ;  15°.3  ;  40°.7  ;  72°.5  ;  0°.0  ;  0°.0  ;  1°.3. 

127.  If  instead  of  considering  the  cooling  of  a  cylinder  as  in  Art.  123  we 
have  to  deal  with  a  cylindrical  shell  whose  curved  surfaces  are  co-axial 
cylinders,  we  are  obliged  to  use  the  Bessel's  Functions  of  the  second  kind. 
Let  our  equations  of  condition  be 

M 1=  0     when     s  =  0 , 
u=f(r)     ''         z  =  b,     . 

Then  (v.  Art.  123) 

V  =  sinh  {fx,z)^J,(i,,r)  -  1^  Al,(/.,r)] 


u  =  0 

when 

r  =  a, 

u  =  0 

" 

r^c. 

Chap.  VIT.]  CYLINDRICAL   SHELL.  231 

where  jx,.  is  a  root  of  the  equation 

^o(y^«)-^^A'-o(;^«)=0  (1) 

will  satisfy  Laplace's  Equation  [(1)  Art.  123]  and  all  of   the  equations  of 
condition  except  the  second. 

is  the  required  solution  if 

/(,.)  J^A,  [,_^.(^„,)  _^^  A-.(^,-)].  (3) 

The  development  (3)  is  easily  obtained. 

Call  the  parenthesis  for  the  sake  of  brevity  B^^ix^r).     Then  by  the  method 
of  Art.  125  we  get  if  we  integrate  over  our  cylindrical  shell 

jrB,{ti,r)B,{jiir)dy  =  0  (4) 

if  [I,,  and  /i,^  are  roots  of  (1) ;  and  by  an  easy  extension  of  (10)  Art.  122 

jrlB,{f,,r)Jdv  =  h{!''W{l^^J  -  <'\B,;(^.,a)J}.  (5) 

Determining  the  coefficients  in  (3)  as  in  Art.  124  and  simplifying  by  the 
aid  of  (4)  we  have 


(6) 


EXAMPLE. 


If  a  membrane  bounded  by  concentric  circles  of  radius  a  and  radius  h,  and 
fastened  at  the-edges,  is  initially  distorted  into  a  form  symmetrical  with  respect 
to  the  centre,  and  then  allowed  to  vibrate 

y  =  ^A,  cos  {^Ji,ct) [ J-„(/.,r)  -  ^^^  K^if^.r)^ 

where  A,,  is  obtained  from  (6)  Art.  127  by  replacing  c  by  h. 


232 


CYLINDRICAL    HARMONICS. 


[Art.  128. 


128.     If  in  the  cooling  of  a  cylinder  u  =  0  when  ;^  =  0,  u  =  0  when  z  =  b, 
and  u  =/(s)  when  r  =  a,  the  problem  is  easily  solved. 

If  in  (2)  and  (3)  Art.  123  fx  is  replaced  by  fii  we  can  readily  obtain 

,v  =  sin(/A«)Jo(/^n) 

and  z  =  cos  (fiz)  t/o(/^*'0 

as  particular  solutions  of  Laplace's  Equation  [(1)  Art.  123]  ;   and 


and  is  real. 


Jo(xi)  _  1  +  ^  ^  2M2  "^  2-. 42. 6^  "*" 


M  =  XAsiJ^ 


where 

by  Art.  31  (7)  and  (8). 

Hence 
is  our  required  solution. 


2  r 


b 
kirz 


sm-—— fZ« 
b 


/kirriX 

V^     .    k'jrz'^'Vir) 
u  =  2^  A„  sm 


b        /k7rai\ 


(1) 


(2) 


(3) 


EXAMPLES. 

1.    If  the  cylinder  is  hollow  and  we  have  u^O  when  z  =  0,  u  =  0  when 
z  =  b,  11  =  0  when  r  =  c,  and  u  ^=f(z)  when  r^=  a  ;  then 


Xa 


kirz 


where  A^.  has  the  value  given  in  (2)  Art.  128,  and 
Ko(xi)  =  Ko(x{)  —  Jo(xi)  log  i 


=  Jo(xi)  log  aj  — ; 


X\  +  h) 


^  / kirc i\       -—  / kircA\ 


(i+i+j)  + 


22  I  2^.4^^  2^.4^.6^ 

[v.  (4)  Art.  120],  and  is  real. 

2.  A  hollow  cylinder  6  feet  long  Avhose  inner  surface  has  the  radius  3  inches, 
and  whose  outer  surface  has  the  radius  1  foot,  has  its  bases  and  outer  surface 
kept  at  the  temperature  0°,  and  its  inner  surface  at  the  temperature  100°,  until 


Chap.  VII.]  TEMPERATURES    UNSYMMETRICAL.  233 

the  permanent  state  of  temperatures  is  established  ;  find  the  temperatures  of 
two  points  in  a  plane  parallel  to  the  bases  and  half-way  between  them,  one  of 
which  is  6  inches  and  the  other  9  inches  from  the  axis.  Ans.,  49°.6;  20°.2. 

129.  If  in  the  problem  of  Art.  123  the  temperatures  of  the  points  of  the 
upper  base  of  the  cylinder  are  unsymmetrical  so  that  u  ^/(r,<^)  when  z  =  b, 
we  have  to  get  particular  solutions  of  Laplace's  Equation  [(1)  Art.  123]  for 
the  case  where  B^u  is  not  equal  to  zero.     We  readily  find  that 

u  =^  sinh  (/jLz)[A  cos  n<f>  +  B  sin  ?i<I>']J^(/jL)-) 

and  u  =  cosh  (ijlz)[A  cos  n(j>  +  B  sin  n<f)]J„(/j,r) 

are  such  solutions,  and  that 

Xx-A  sinh  a,.z  ^  ,  ^        . 

A  sinh  a  b  1^    "•^-  °°'  ""^  +  ^"''  '''"  «^]'^«(^^'')  (1) 

is  the  solution  of  the  given  problem  if 

./(n<A)  =  X    X  ('^^'a- COS  «(^  +  ^«,-i- sin  «(^y„(^,.;-)  (2) 

where  /Aj.  is  a  root  of  the  equation 

EXAMPLES. 

1.  Show  that 

2.  Show  that 

0 


(3) 


234  CYLINDKICAL   HARMONICS.  [Art.  129. 

3.    Show  that  in  Art.  129 

'•)dr 


j'd<t>Jrf(r,<j>)J,(fi, 


^o.k  ■ 

TT             a'^[Ji(/i^,o 

')J 

B,y- 

=  0, 

A 

2k             a 

jd4Jrf{r,4>) 

■"  (1           n 

COS  n(f>J„(ti^. 

r)dr 

A,k  - 

«W  + 

.:(/x.«)]-^ 

Cdcf)  j  rf(r,<j))  sin  n4)J„(ix^r)dr 

4.  Obtain  the  coefficients  for  the  case  where  the  convex  surface  of  the 
cylinder  is  impervious  to  heat. 

5.  Obtain  the  coefficients  for  the  case  where  the  convex  surface  of  the 
cylinder  is  exposed  to  air  at  the  temperature  zero. 

6.  Show  that  if  in  a  drumhead  problem  of  Art.  11  the  initial  distortion  is 
unsymmetrical,  so  that  we  have  to  solve  the  equation  [xi]  Art.  1  subject  to 
the  conditions  z  =f(i\  <^)  when  ^  =  0,  Z>,,~  =  0  when  f  =  0.z  =  0  when  r  =  a, 
the  solution  is 

^  =  X    X  ^^^  ^^'''*^ ^^"''  ^°^  '^*^  "^  ^"' "  ^^^  "'^)'^«(^'^'') 

where  A^^^,  B^^^,  A^^,.,  and  B„j,  have  the  values  given  in  Ex.  3. 

7.  What  modifications  do  the  statements  made  in  Ex.  2,  Art.  126,  need  to 
make  them  apply  to  the  unsymmetrical  case  treated  in  Ex.  6  ? 

Show  that  any  possible  nodal  system  in  Ex.  6  is  composed  of  concentric 
circles  and  of  radii  whose  outer  extremities  are  equidistant,  v.  Eayleigh's 
Sound,  Vol.  I.,  Arts.  (202-207). 

8.  Solve  the  problem  of  Art.  127  and  of  Art.  127,  Ex.  for  the  unsym- 
metrical case.  Suggest  ion:  AJ„(x)  +  BK„(x)  is  a  solution  of  Bessel's 
Equation. 

9.  Solve  the  problem  of  Art.  128  and  of  Art.  128,  Ex.  1,  for  the  case  where 
u=f{z,<f)  whenr  =  a.  Suggestion:  71  =  sin  fiz(A  cos  7i<l> -{- B  sin  7i(f>)Jj,(fji7-i) 
is  a  solution  of  Laplace's  Equation,  and  /'(.:;.  cf})  can  be  developed  into  a  double 
Fourier's  Series  [v.  (15)  Art.  71]. 


Chap.  VII.]  EXAMPLES.  235 

10.  Show  that  in  dealing  with  a  wedge  cut  from  a  cylinder  by  planes 
passed  through  the  axis,  or  with  a  membrane  in  the  form  of  a  circular  sector, 
it  may  be  necessary  to  use  Bessel's  Functions  of  fractional  or  incommensurable 
orders. 

11.  BernouillV s  Problem  (v.  Chapter  IX).  In  considering  small  transverse 
vibrations  of  a  uniform,  heavy,  flexible,  inelastic  string  fastened  at  one  end 
and  initially  distorted  into  some  given  curve,  we  have  to  solve  the  equation 
Diij-=(P'{xB^y -\- D^li),  subject  to  the  conditions  I>^y^=^  when  if  =  0, 
y  =/(.x)  when  i!  =  0,  ?/  ^  0  when  .r  ^  a  ;  the  origin  being  taken  at  the 
distance  a  below  the  point  of  suspension  and  the  axis  of  X  taken  vertical. 

Show  that  y'^Zj  ^'^k  ^^'^  ^'^'^-^  Ai(/^A-^)  1 

k=  1 

where  B^{x)  =  l-~-\-  ^^  -  py|^,  +  •  •  ' 

='/o(2v':^) 

and  /i^.  is  a  root  of  the  equation 


^fix)  B,{ix^  x)dx       jf(x)  J,{2  fji,  sl~x)dx 


12.  As  a  simple  case  under  Example  10  consider  the  vibrations  of  a  circular 
membrane  fastened  at  the  perimeter  and  also  along  a  radius  and  then  initially 
distorted  (v.  Rayleigh's  Sound,  Art.  207).  In  this  case  we  must  modify  the 
formula  given  in  Ex.  6  by  dropping  out  the  terms  involving  cos  w^  and  by 

taking  •/<  =  — •     The  required  solution  is 


= X  X  ^'"'^  ^°^  ^^*^^  ^^^  2  "^-(^'^■'■^ 


where  /w^,  is  a  root  of  ^^^  ^^^   =  0 


277  a 

^^  ^d<\>^rf{v,^)  sin^J-^,(;U,r)cZ7 


and  ^--^-TT  ct}{,J.!{ix,a)J 

2 


236  CYLINDRICAL    HARMONICS.  [Akt.  129. 

For  the  terms  in  which  m  is  odd,  Jm(x)  can  be  readily  obtained  from  (13) 

2 

Art.  122,  whicli  will  become  a  finite  sum. 
For  exami)le,  (13)  Art.  122  gives  the  values 

M^)  =  -  \  d  [  (l  +  .^)  «i^^  "^  +  1  cos  x~j  ;     &c. 

13.  The  question  of  the  flow  of  heat  in  three  dimensions  involves  a  problem 
not  unlike  the  last. 

Suppose  the  initial  temperatures  of  all  points  in  a  sphere  of  radius  c  given, 
and  let  the  surface  be  kept  at  the  temperature  zero.  Then  Ave  have  to  solve 
the  equation 

7>,«  =^;[a(-A.O  +^,I'.(sin^D.«)  +-^,^l«]  (1) 

([iv]  Art.  1)  subject  to  the  conditions 

7/^0     Avhen     r  =  c, 
u  =/(>;  6,  <t>)     when     f  =  0. 

If  we  assume  u  —  T.B.  V  where  T  is  a  function  of  t  onlj^,  B  of  ;•  only,  and  V 
of  6  and  <^  only,  (1)  can  be  broken  up  into 

'^+u'a^T=0  (2) 

m{m.  +  1)  r+  ^  A(sin  OD^  V)  +  ^  D^  V=  0  (3) 


(4) 


Hence  T=  e" "'"=',   V=  Y^(fi,  <f>)  [v.  Art.  102  (2)],  and  B  is  still  to  be  found. 
If  in  (4)  we  let  x  =  ar  and  z  =  Rsfar  it  becomes 

which  is  satisfied  by  z=^J^^^(x).     (v.  Art.  17.) 

Therefore  ^  =  "7=  '^m  +  i(«»*)- 

ya?' 


Chap.   \U.]  FLOW    OF    HEAT    IN    A    SPHERE.  237 

f(r,  0,cf>)=^j^^     Cd4,  Cf(r,  $1 ,  </,i)P,„(cos  y)  sin  OidO,  by  (3)  Art.  114, 

=  X    X^"^---^--"^''^  '''''  ''"^  +  ^»,,r,K,A^)  sin  ^<^]P,:i(/x). 

m=0    n=(l 

1=  -X 

where  a^  is  a  root  of  the  equation 

and  C„,  „  ,  =  -^ — — 

k-  =  » 

i-=  1 
where  D^_„_^  =  -i^ 


The  final  solution  is 


cf.  Riemann,  Par.  Dif.  Gl.,  §§  72  and  73. 


CHAPTER    VIII. 

Laplace's  equation  in  curvilinear  coordinates, 
ellipsoidal  harmonics. 


130.      Ortliorjonal  Curvilinear  Coordifiates. 
If 


F,(x 

Vj^)' 

=  pi 

F,{x 

Ih  -)  -- 

=  P2 

F,(x 

y^^) 

=  P3 

^) 


are  the  equations  in  rectangular  coordinates  of  three  surfaces  that  are  mutually 
perpendicular  no  matter  what  the  values  of  pi,  p2,  and  ps,  the  parameters  p^, 
P2,  and  ps,  may  be  regarded  as  a  set  of  coordinates  for  a  point  of  intersection 
of  the  three  surfaces,  in  the  sense  that  when  pi,  p^,  pz  are  given  the  point  in 
question  is  determined,  and  when  the  point  is  given  the  corresponding  values 
of  pi,  p2,  Ps,  can  be  found. 

Prom  equations  (1)  x,  y,  and  z  can  be  expressed  in  terms  of  pi,  p^,  and  p^. 
Suppose  this  done.  If  now  x,  y,  z  are  the  rectangular  coordinates  of  the 
point  pi  =  o,  p-i  =  ^,  Ps  =  c,  the  rectangular  coordinates  of  the  points 
p^  =  a  +  rfpi,  p2  =  h,  p^  =  e,  are  obviously  x  +  BpX.dp^  +  ci,  ?/  +  Dp^y.dpi  +  ^2, 
z-\-D  z.dpi-\-es,  where  £1,  co?  and  cg  are  infinitesimals  of  higher  order  than 
dpi .  Hence  the  square  of  the  distance  between  the  points  will  differ  by  an 
infinitesimal  of  higher  order  than  that  of  dp{  from  dn^  where 

Let  ^^  =  (D,xf  +  (lJ,^yy  +  (.Z>,,.)^ 

L  =  (D,xr+(n,j/f+(I),^f 

Y^={F>,xy+{Dpjjy^{D,^z)\ 


(2) 


Then  if  dui  is  the  element  of   length  normal  to  the  surface    pi  =  a,    dn^ 
normal  to  po  =  h,  and  dn^  normal  to  pg  =  <? 

dni='^,      ■  dn2  =  '^,         dn,  =  '^.  (3) 


CURVILINEAR    COORDINATES.  239 

The  element  of  surface  diS^  on  the  surface  pi^  <i  is  easily  seen  to  be 

.j^3,  (4) 


and  the  element  of  volume  do  is 


du  =  '^^'' 

EXAMPLE. 

Show  that  hi  =  {D,p,y  +  {Dyp.y  +  (Api) 

hl  =  {D^.p,Y  +  {D,p,y-^{D.^p,y 
hi  =  {D,p,y  +  {D,jp,y  +  (Apb)^. 

Suggestion:    If   7ii  has  the  value  just  given     — ^S    — f^,    ^^     are  the 

il\  111  ll^ 

direction  cosines  of  the  normal  at  any  given  point  of  p^  =  a.     (v.  Int.  Cal. 
page  161.)     Then 

131.     Laplace's  Eqiiation  in  orthogonal  curvilinear  coordinates. 
If  we  apply  the  special  form  of  Green's  Theorem 

fff^''  Vdxdydz  =   Cb,  VdS  (V.  Art.  98) 

to  the  space  bounded  by  the  surfaces    pi^a,    p.2^b,    pz^=c,    p^^  a.  -{-dp^, 
P2  =  f'-{-  (^P2 1    ps  =  ^  +  dpz  J    we  have 

whence 

v=  v=  /v./-,[i>„ {^D,, r)  +  D,,(j!^D,, r-)  +  D,,iJ^D,, r)] ,  (6) 

and  Laplace's  Equation  in  our  curvilinear  system  is 


240  ELLIPSOIDAL    HARMONICS.  [Art.  132. 

If  it  happens  that  V"/3i  =  **'  ^  ==  pi  will  satisfy  (7)  and  we  shall  have 
h,hJi,Dj^\=^0.  In  like  manner  if  V-p.  =  0  we  have  dJj^\=0, 
and  if  V-/03  =  0  we  have  Dp/— y- )  :=  0  ;  and  therefore  (7)  reduces  to 

hfir-^  F+  hW^V^  hlD^J=  0  (8) 

when  VVi  =  0,  VV2  =  0,  and  Ws  ==  0. 

132.  If  instead  of  having  the  value  of  the  Potential  Function  V  given  on 
the  surface  of  a  sphere  as  in  our  Spherical  Harmonic  problem,  we  have  it 
given  at  all  the  points  on  the  surface  of  an  oblate  spheroid,  and  are  required  to 
find  its  value  at  any  internal  or  external  point,  we  can  easily  get  a  solution  by 
methods  in  no  essential  respect  different  from  those  already  employed,  if  only 
we  rightly  choose  our  system  of  coordinates. 

If  we  take  an  ellipse  and  an  hyperbola  having  the  same  foci,  and  revolve 
them  about  the  minor  axis  of  the  ellipse,  we  shall  get  a  pair  of  surfaces  which 
are  mutually  perpendicular  ;  a  plane  through  the  axis  of  revolution  will  cut 
both  the  spheroid  and  the  hyperholoid  orthogonally. 

The  equations  of  the  three  surfaces  can  be  written  :  — 

i?,(;r,j,,«,A)=^  +  ;^,  +  |;-l=0  (1) 

^(.,2,,,,M)=^  +  ^  +  i;-l=0  (2) 

F^(x,  y,  z,  v)=z  —  vx=^0,  (3) 

where  X^>  ^^>  yu-^,  2  b  being  the  distance  between  the  foci. 

For  all  values  of  A,  fi,  and  v  consistent  with  the  inequality  above  written 
the  surfaces  (1),  (2),  (3)  intersect  in  real  points  and  cut  orthogonally. 

X,  fM,  and  V  can  be  so  chosen  that  the  surfaces  will  intersect  in  any  given 
point,  and  therefore  can  be  taken  as  a  set  of  curvilinear  coordinates,  and 
Laplace's  Equation  can  be  expressed  in  terms  of  them  by  the  aid  of  Formula 
[xv]  Art.  1. 

From  (1),  (2),  and  (3)  we  readily  get 


b%l  +  v') 
i)  7.2 


xV'i'' 


^»2(l  +  v'') 


(4) 


Chap.   VIII.]  SPHEROIDAL    COORDINATES.  241 


u.  A     j/i'^  —  u,^  av 

whence  D),x  =     .  >  D^y  =  7  \  7:^ tt  »  D^z  =  -—  ; 

h^l  +  v''  b^k--b'  b\ll  +  v^ 

and  A?  =  X^^T^  (^> 


[v.  130  (2)].     In  like  manner  we  get 


1  _A^-/i^ 


(6) 


and  ^^  =  ^^(1  +  .-^)-^'  ^^> 

and  [xv]  Art.  1  becomes 

which  is  Laplace's  Equation  in  terms  of  our  SpJieroidal  Coordinates  \,  fi,  and  v. 

If  now  in  place  of  A,  ^l,  and  v  we  can  introduce  some  function  of  A,  some 
function  of  fi,  and  some  function  of  v  which,  therefore,  will  represent  the 
same  set  of  orthogonal  surfaces,  and  if  we  can  choose  three  functions  a,  p, 
and  y,  which  of  course  are  functions  of  x,  y,  and  z,  so  that  V^a  =  0, 
V^/8  =  0,  and  W  =^  0,  equation  (8)  must  reduce  to  the  simple  and  sym- 
metrical form  given  in  [xvi]  Art.  1. 

These  functions  a,  (3,  and  y  are  easily  found.  Equation  (8)  is  V'T— 0 
expressed  in  terms  of  A,  /x,  and  v.  Assume  that  F  is  a  function  of  A  only  ; 
then  D^V=0,  and  i),F=0,  and  (8)  reduces  to 


A[AN/A--/>2.Ar]  =  o 

whence 

K<J\'-P 

and 

v=i^^-'\, 

and  is  a 

function  of  A 

which  satisfies  Laplace's  Equation. 

242  ELLIPSOIDAL    HARMONICS.  [Akt.  132. 

Take  this  as  a  leaving  e^  at  present  undetermined,  so  that 

rfa  =  — ,  and     a^  — see    '-• 

aVa-  -  h-  (>         i> 

In  the  same  way  we  get 

./^  =  -M^,    and    /3--^^sech-^, 

(V.  Int.  Cal.  Art.  46,  Ex.) 

dy  =  3-1 — :,'    and     y  =  ('3  tan"^  v. 
'       1  -\-  V- 

Substituting   these  vahies  in  (8)  and  taking    (\  =  —  r^  =  I>,    and    (-3  =  1, 
(8)  reduces  at  once  to 

or  since  X  ==  i  sec  a ,     fi  =^  b  sech  /3 ,     and     v  =  tan  y ,  (10) 

to  cos^  a  D;  V  +  cosh^  jB  D/  T  +  (cosh'^  yS  —  cos'-^  a)Z>^'  T  ==  0  (11) 

which  is  Laplace's  Equation  in  terms  of  what  we  may  call  Normal  Oblate 
Spheroidal  Coordinates. 

In  using  (11)  it  is  to  be  noted  that  the  point  whose  coordinates  are  (a,  /3,  y) 
is  the  point  of  intersection  of  an  oblate  spheroid  whose  semi-axes  are  b  sec  a 
and  b  tan  a,  an  unparted  hyperboloid  of  revolution  whose  semi-axes  are 
b  sech  ^  and  b  tanh  )8,  and  a  plane  containing  the  axis  of  the  system  and 
making  the  angle  y  with  a  fixed  plane  ;  and  that  if  the  axis  of  revolution  is 
the  axis  of  Y  and  the  fixed  plane  is  the  plane  of  XY,  the  rectangular  coordi- 
nates of  (a,  (i,  y)  are 

x^^b  sec  a  sech  /3  cos  y,   ij^=b  tan  a  tanh  (i,  z^=b  sec  a  sech  j3  sin  y       (12) 

[V.  (4)]. 

If  now  we  let  a  range  from  0  to  — ,  fB  from  -co  to  00  ,  and  y  from  0  to  27r, 

we  shall  be  able  to  represent  all  points  in  space  ;  and  if  we  agree  that  negative 
values  of  /3  shall  belong  to  points  below  a  plane  through  the  origin  and 
perpendicular  to  the  axis  of  revolution  and  positive  values  of  /3  to  points 
above  that  plane,  not  only  shall  we  have  no  ambiguity,  but  also  the  rectangular 
coordinates  of  any  point  as  given  in  (12)  will  have  their  proper  signs. 


Chap.  VIII.]  SPHEROIDAL   COORDINATES.  243 

EXAMPLES. 

1.  If  the  spheroid  is  a  prolate  spheroid,  the  ellipse  and  confocal  hyperbola 
must  be  revolved  about  the  major  axis  of  the  ellipse,  and  the  plane  must  con- 
tain that  axis.  In  place  of  equations  (1),  (2),  and  (3)  of  Art.  132  we  have, 
then, 


f;+^+^-i=o 

fi'       fi~  —  Ir       fi-  —  ir 

z  —  vij  =  0 

K'>J/>  fi\ 

\^ 

-b'         ifi-fx'              h%i  +  vy- 

X-- 

-f,^'     ■'•'        A— /x^'     "'       {X'-b'){lr-fx') 

where 


Laplace's  Equation  becomes 

P^  A[(x- ^■^)Ar] +p^ />,[(.'- ^=)/),  n 

+  ^X?^=^A.[(i+.')Z).F]  =  o.  (1) 

(l),educesto     ^+ ^+ ^^  ^-^_  ^^D^V  =^,  (2) 

where  da  =  —  — -,  >     d(3  =  7:^ — '—-;,  >     d\  — 


X'  —  b''        '        b'  —  iJi''  l  +  y'^ 

a  =  ctnh~  ^  V  >     /3  =  tanh~ '  ^  ?     and     y  =:  tan"  ^  v . 
b  b 

Since  X^  b  ctnh  a,     /n  =  (*>  tanh  (3,     and     v  =  tan  y 

(2)  can  be  reduced  to 

sinh^a  Z>,'  V-\-  cosh^  (3D^-V-\-  (sinh-'  a  +  cosh^  ^)i)/  P^=  0.  (3) 

In  using  (3)  it  is  to  be  noted  that  the  point  (a,  ft,  y)  is  the  point  of  inter- 
section of  a  prolate  spheroid  whose  semi-axes  are  b  ctnh  a  and  b  csch  a,  a 
biparted  hyperboloid  of  revolution  whose  semi-axes  are  b  tanh  (3  and  b  sech  /3, 
and  a  plane  containing  the  axis  of  revolution  and  making  the  angle  y  with  a 
fixed  plane. 


244  ELLIPSOIDAL   HARMONICS.  [Akt.  133. 

If  the  fixed  plane  is  that  of  (AT)  the  rectangular  coordinates  of  any  point 
(a,  /3,  y)  are 

X  —  6  ctnh  a  tanh  (3,    ij  =  b  csch  a  sech  ^  cos  y ,    z  —  h  csch  a  sech  /3  sin  y , 

and  a  may  range  from  oo  to  0,  yS  from  —  oo  to  oo ,  and  y  from  0  to  'Itt. 
Negative  values  of  /3  are  to  be  taken  for  points  lying  to  the  left  of  a  plane 
through  the  origin  perpendicular  to  the  axis  of  revolution. 

2.  Transform  Laplace's  Equation  in  Spherical  Coordinates  [xiii]  Art.  1 
to  the  symmetrical  form 

a- D: V  +  cosh2 ^ j^'i Y  ^  ^osh^ ^ B^ F  =  0 

1  (9 

where  *  — ",'     yS  =  logtan-)     and     y  =  <^. 

3.  Transform  Laplace's  Equation  in  Cylindrical  Coordinates  [xiv]  Art.  1 
to  the  symmetrical  form 

Bl  V^BIV^  e'-B;  r  =  0 

where  a  =  log  r ,     fi=<i>,     and     y  =  .t . 

133.  In  each  of  the  cases  we  have  considered,  it  has  been  easy  to  pass 
from  Laplace's  Equation  in  terms  of  the  chosen  coordinates  representing  an 
orthogonal  system  of  surfaces  to  the  symmetrical  form  [xvi]  Art,  1 ;  and  it  is 
evident  that  our  new  coordinate  a  is  a  value  of  V  corresponding  to  such  a 
distribution  that  the  surfaces  obtained  by  giving  particular  values  to  p^  are 
equipotential  surfaces ;  that  y8  is  a  value  of  V  corresponding  to  such  a 
distribution  that  the  surfaces  obtained  by  giving  particular  values  to  p.,  are 
equipotential  surfaces ;  and  that  y  is  a  value  of  V  corresponding  to  such  a 
distribution  that  the  surfaces  obtained  by  giving  particular  values  to  p^  are 
equipotential  svirfaces.  a,  /3,  and  y  are  called  by  Lame  <'  thermometric 
parameters.'''' 

The  condition  that  these  values  should  exist,  for  a  given  system  of  surfaces, 
that  is,  that  the  distribution  described  above  should  be  possible,  is  readily 
obtained.  We  shall  work  it  out  for  a.  It  is  merely  the  condition  that  V  in 
Laplace's  Equation  may  be  a  function  of  p^  alone. 

If   F  is  a  function  of  pi  alone 

i>.y="^i>,P.,  D,v=%D„,,  ^'.^•=';^'ap.. 


Chap.  VIII.] 


THERMOMETRIC    PARAMETERS. 


245 


D,fV  = 


dpi 


(AjpO'-h 


dV 


vPi 


DJ'V: 


dp^  dpi 


Therefore  [(D^p^y  +  (D,,p,y  +  (I),p,y-]  '|^V  [D^p,  +  DJ-p,  +  D^  '^=  0 

(ipi  dpi 


whence 


(D,p,y  +  (D^p,y+(D,p,y 


d^V  .  (IV 
dpj;    '   dpi 


where  i^i(pi)  may  be  any  function  of  p^  alone.      Uur  required  conditions  are 
then 

-Jf-^^(pO 


--r  —  F,(p.^ 


—TV  —  ^3(^3) 


(1) 


and  when  they  are  fulfilled  the  original  curvilinear  coordinates  p^,  p^,  ps, 
correspond  to  possible  equijjotential  or  isothermal  surfaces,  ther  mo  metric 
jjarameters  a,  ^,  and  y  exist,  and  the  reduction  of  Laplace's  Equation  to  the 
symmetrical  form  [xvi]  Art.  1  is  possible. 

134.  Returning  to  our  Oblate  Spheroid  problem  of  Art.  132  we  can  proceed 
as  usual  to  break  up  our  equation  (11)  Art.  132. 

Assume  that  V=  L.M.N,  where  X  is  a  function  of  a  only,  J/ of  /3  only, 
and  A^of  y  only.      (11)  Art.  132  becomes 

cos^g  d^L      cosh^  (3  d^3I       [cosh^  (S  —  cos^  a]  d^N _ 
L     da^'^      M      dfi'^  '  N  ^  ~  ^ 


dV.       1^ 


cosh2  ^        dm 


1  COS^  0 

Lcosh^fi  —  cos'^  a  da^  '   Jfcosh^^S  —  cos^a 


1  d^JY 

JV  dy'  ' 


The  first  member  is  independent  of  y,  and  the  second  member  is  independent 
of  a  and  /?,  and  the  two  members  are  identically  equal.  The  second  member 
is  then  independent  of  a,  /?,  and  y  and  must  be  constant;  call  it  n^.  We  have, 
then, 

^  +  .W=0  (1) 


246  ELLIPSOIDAL   HAKMONICS.  [Art.  134. 

(1)  gives  us  N=  A  cos  ny  +  i?  sin  wy.  (3) 

(2)  can  be  written 

~T~  "^  ~^  " "       ^  ^ '  ^^^ JT'  ^  ^  ^^^^'^     ^' 

d'^L 
whence  cos^a  -—^  +  ['^^  c'^^" "  ~~  "K"^  +  1)]^  =  0  (4) 

and  cosh2  ^  ^+  [m(m  +  1)  —  n^  cosli^  /3]ilf  =  0.  (5) 

If  we  introduce  x  =  tanli  ^8  in  (5)  it  becomes 

(1  -  -')  "'-  2-  f  +  [»(»'  +  1)  -  I^J  ^^=  0  (6) 

where  since  x  =  tanh  /?  and  fi  may  have  any  value  from  —  oo  to  oo  ,  a;  may 
have  any  value  between  —  1  and  1.  (6)  is  a  familiar  equation  having  for  a 
particular  solution 

M=  (1  -  xy  '^-^1^  =  PZ(x)  =  P^(tanh  (S).  (7) 

(V.  Arts.  101  and  102) 

If  we  introduce  in  (4)  x  =  tan  a  it  reduces  to 

(l+.,.)g  +  2.g+[j^,-,„(»+l)]i  =  0.  (8) 

(8)  is  an  unfamiliar  equation,  but  it  can  be  treated  as  (6)  was  treated  if  we 
take  the  pains  to  go  back  to  the  beginning  and  follow  the  steps  of  the  treat- 
ment of  Legendre's  Equation. 

V 
This  labor  can  be  saved,  however,  by  noting  that  if  we  let  x  =  "4  (8)  becomes 

and  is  identical  in  form  with  (6).     Hence 

L  =  P»(^)     and     L  =  (l-  f)!  '^-^f^  (v.  Art.  101), 

where  y  =  i  tan  a,  are  particular  solutions  of  (4). 
AVe  can  avoid  imaginaries  if  we  use  the  values 

L  =  (-i)"'-"P:,\Q/)      and      L  =  P"  + "  +  \1  -  y^)!  ^^"^"f'^^-  (9) 

ay 


Chap.  VIII.]  SPHEROIDAL    HARMONICS.  247 

Since  we  assumed    V=  L. M. N  we  have 

V  —  {A  cos  ny  +  B  sin  7iy)P,;;(tanh  fi)  (—  /)'«  -  "P,',;(i  tan  a)  ^ 

and         r=  (.4  cos  «y  +  B  sin  7^7)P,«(tanh  /3)  /'"  +  "  +  ^  sec"  a  ^^"^^';»(^^ana)   V  (10) 

^  ^  ^^    '"^  '^^  (f/(aana))«  J 

as  particular  solutions  of  (11)  Art.  132. 

If  the  problem  is  symmetrical  with  respect  to  the  axis  of  the  spheroid 
D-V=^(),  7^2  =  0  and  our  particular  solutions  (10)  reduce  to 

r=  (-  0'«P,„(/  tan  a)P^(tanh  ^)  | 
and  r= /'«  +  !(),„(*■  tan  a) P„,(tanh^).    J  ^ 

If,  then,  V  is  given  on  the  surface  of  a  spheroid  as  a  function  of  fi  and  y, 
we  must  express  it  as  a  function  of  tanh  ji  and  y,  and  shall  be  obliged  to 
develop  it  in  terms  of  Spherical  Harmonics  of  tanh  y8  and  y  by  the  formulas  of 
Chapter  VII,  using  the  first  equation  in  (10)  for  the  value  of  V  at  an  internal 
point,  and  the  second  for  the  value  of  V  at  an  external  point.  If  the  problem 
is  symmetrical,  we  must  develop  in  Zonal  Harmonics  of  tanh  (i  by  the  formulas 
of  Chapter  VI. 

A  convenient  form  for   Q^(i  tan  a)  is  obtained  from  (2)  Art.  100  ;  it  is 

/dx 
Hence  ^o(*  tan  a)=  —  ii  ^  ^^^  =  —  iY|  —  aj-  (13) 

tan  a 

EXAMPLES. 

1.  A  conductor  in  the  form  of  an  oblate  spheroid  whose  semi-axes  are 
b  sec  ao  and  h  tan  a^  is  charged  with  electricity  and  is  found  to  be  at  potential 
Vq  ;    find  the  value  of  the  potential  function  at  any  internal  or  external  point. 

Here    Fo=  FoPo(tanh /3).     Hence  at  an  internal  point  , 

and  at  an  external  point 

^o(aana)  _       \2~  V 

^       '^«a(*tana,)^"^*'^^^^^-^77r_     >.  ^^^ 

\2      V 

Since  V  in  (2)  involves  a  only,  the  equipotential  surfaces  are  all  spheroids 
confocal  with  the  conductor. 


248  ELLIPSOIDAL   HARMONICS.  [Art.  135. 

2.  The  upper  half  of  an  oblate  spheroid  whose  semi-axes  are  h  sec  a^  and 
b  tan  ao  is  kept  at  the  temperature  unity,  and  the  lower  half  at  the  tempera- 
ture zero.     Find  the  permanent  temperature  at  any  internal  point. 

1  ,   3Pi(itana)  „  ^^     .    .,,       7  lP3(itana)              ^   a\    \ 
A71S.     u  =  ^  +  7  TTT^T T  Pi(tanh  /S)  —  -  •  -  ^f— ~  P3(tanh  p) -\ 

2  4Pi(ttanao)     ^  8  2P3(itanao)       ^ 

(v.  Art.  93).  u  may  be  expressed  in  terms  of  x,  y,  and  z  without  serious 
difficulty  [v.  (12)  Art.  132]. 

_1   ,   3//_7  1  1  \1hif  -  \^y{x'  +  if  +  ^ -  h^)  -  O^'V]   , 
"~2^4c      8'2'2  hc^-^Wc  "^ 

if  2c  ^  lb  tan  a^  =^  minor  axis  of  spheroid. 

135.  Let  us  now  find  the  potential  function  at  an  external  point  due  to 
the  attraction  of  a  solid  homogeneous  oblate  spheroid,  using  the  method  em- 
ployed in  Arts.  98  and  99. 

Consider  first  the  potential  function  due  to  a  shell  bounded  by  the  spheroids 
for  which  a^=  <f)  and  a  =  (f>-\-  dcj). 

By  (1)  Art.  98  we  have 

4npK^[I)„V,-D„r,-],^^,  (1) 

where  p  is  the  density  and  k  the  thickness  of  the  shell,  Fj  the  value  of  the 
potential  function  at  an  internal  point,  and  Vo  the  value  of  the  potential 
function  at  an  external  point. 

Let  Fi  =  ^A,J-  0'«P„.(i  tan  a)P,„(tanh  /5) 

and  F.  =  ^B^J^'  +  'Q^il  tan  a)P,„(tanh  /?)       [v.  (11)  Art.  134]. 

Since  Vi  and  Vo  must  have  the  same  value  when  a^=  (f> 

^-      ^™^         P.(/tanc^)       ^     '^  ''-J(l+x%F^(xi)f  ^'> 

tan  <p 

[V.  (12)  Art.  134]. 

Hence      V,  =  ^t'«P^P^(tanh /3)P„,(i  tan  a)j  ^^  ^  .^.^^p^^_ 


and  V,  =  5)i'«P^P,„(tanh  (3)P^(i  tan  a)f—-^ 


xi)J 
dx 


^^^^Jl-\-x%F„,(xi)Y 
X>„Fi  =  P»„Fi.P»„a.  D„V^  =  D,V^.D^a 


(3) 


Chap.  VIIL]  ATTBACTION   OF  A   SPHEROID.  249 

tan  a 

V,  -  n  =  Xi"B..  p..  (tauh  «  P,„  (,•  tan  a)f  .^^  . 

tan<|)  ^ 


(Xt 


)J 


(/P„,(/tana)      /^  dx  n 

d7i  =  '^  =  '-^  =  '^^'  ~  ^'dX  =  bseca  Vtan^a  +  tanh'-^^S.fZa  (4) 

V.  Art.  130  (3),  and  Art.  132  (5)  and  (10). 

1 


[A,a]a 


0  sec  <^  Vtan^  <^  +  tanh^  ^ 


Hence    [  A,  F,  -  Z)„  n]„  _  ,  = — -^^ Xi^B^-'f.f'^^ 

■-  «      J      *      ^secc^N/tan^  +  tanh^^^        "'P,„(itan<^) 

K  =  [(//^]„  ^  ^  =  /;  sec  </)  Vtan2<^  +  tanh^/3 .  (7<^ 
by  (4),  and  (1)  may  be  written 


•  i.pP  sec^c^(tan^<^  +  tanh^/3)./<^  =  ^^'"'^'"plcftan^)' 
Since  tanlr'  /3  =  -^  Po(tanh  /3)  +  §  Po(tanh  fS) 

l)y  (5)  Art.  95,  to  satisfy  (5)  we  must  give  m  the  values  0  and  2  and 

Po  =  -J Trp/r  sec^  c/)(3  tan^  c^  +  l)rf</> 
and  p2  =  ^  ^pf^^  sec^  c^(3  tan-  </.  +  1)(Z<^ . 


(5) 


250  ELLIPSOIDAL    HARMONICS.  [Akt.  135. 

So  that  by  (3) 

F,.  =  I  Trpb^  sec^  <^(3  tan^  «^  +  l)./*^!"    (^^—2 

tan  (|) 


'(^""^^)^'(''*""')/(i+.^)[f.(.w)]-]     (•') 


and        Vo  —  ±  7rp6-  sec'^  ^(3  tau"-^  c^  +  l)(lcf)[iQQ(i  tan  a) 

+  f^P.Ctanli  /8)  ^2(i  tan  a)] .        (7) 

The  potential  function  at  an  external  point  due  to  the  solid  spheroid  for 
which  a  =  tto  is 

V—  i  Vn  —  I Trpb^  sec^  tto  tan  ao[iQo(i  tan  a)  +  I'^PoCtanh  /3)  (),(i  tan  a)].  (8) 

If  2a  is  the  major  axis  and  2c  the  minor  axis  of  the  spheroid 

,      7  0       o      ^  ,Trpa^c      M 

^trplr  sec-  tto  tan  a,,  =  ^  -^^^ —  =  — 

where  ili"  is  the  mass  of  the  spheroid.     Therefore 

M 

V=  J  \_iQo(i  tan  a)  +  ^^^AC^anh  (S)  Q,{1  tan  a)]  (9) 

is  the  required  value.     (9)  can  be  reduced  to 

F  ==  ^  1 1  -  a  + 1  [  (~  -  «)  (3  tan'^  a  +  1)  -  3  tan  a~\  [3  tanh^  /?  -  1]  }  •     (10) 

EXAMPLES. 

1.  Break  up  the  equation  (3)  Ex.  1,  Art.  132,  for  the  prolate  spheroid,  and 
obtain  particular  solutions  of  the  term 

F=  (A  cos  ny  +  B  sin  «y)P,',;(tanh  /3)P,",(ctnh  a), 
F=  (^  cos  «y  +  B  sin  ny)P,;;(tanh  /3)(-  1)'  cscli"  a'^^|^^^^. 

2.  Break  up  and  solve  the  equations  of  Exs.  2  and  3,  Art.  132,  and  show 
that  they  lead  to  familiar  forms. 

3.  If  in  Ex.  1,  Art.  132,  the  conductor  is  a  prolate  spheroid  whose  semi- 
axes  are6ctnhao  and  Resell ao  show  that 

F=  Vo  at  an  internal  point.  V=^  V^ —  at  an  external  point. 

da 


Chap.  VIII.]  ELLIPSOIDAL    COORDINATES.  251 

4.    Show  that  the  potential  fuuction  at  an  external  point  due  to  the  attrac- 
tion of  a  homogeneous  solid  prolate  spheroid  is 

r=  y  [ ^o(ctnh  a)  —  P^Ctanh  /S)  (^^(ctnh  a)] . 

Ellijpsoldal  Harmonics. 

136.     If  we  are  dealing  with  an  ellipsoid  instead  of  a  spheroid,  we  can  tak^ 
as  our  orthogonal  system  of  surfaces  a  set  of  confocal  quadrics  ; 


X'^X' 

jf_ 

U'  ^  A- 

-1 

=  0 

11 

-^'+1^- 

-1 

=  0 

^"+   y~  +    ^'  -1=0 


(1) 


where  A->  c- >/«-->  ^'>  i^-.  Here  the  first  surface  is  an  ellipsoid,  the 
second  an  unparted  hyperboloid,  and  the  third  a  biparted  hyperboloid.  Each 
of  the  three  principal  sections  of  the  system  consists  of  confocal  conies,  and  it 
is  well  known  and  is  easily  shown  that  the  surfaces  cut  orthogonally.  A,  /i, 
and  V  will  be  our  curvilinear  coordinates,  and  are  known  as  Ellvpsoidal 
Coordinates. 
We  find  without  difficulty  that 

m^        ^  b\c'  —  b^)  "  c\c^  —  l/')  ^'^^ 

(X^-J^^XX^-c^  (f^-PXc'-f^)  (P-v^(^-r/^) 

To  avoid  ambiguity,  we  shall  suppose  that  of  the  nine  semi-axes  in  (1) 
Vc^  —  fi^  is  to  be  taken  with  the  positive  sign  for  a  point  on  the  half  of  the 
unparted  hyperboloid  on  which  z  is  positive,  and  with  the  negative  sign  for  a 
point  on  the  half  on  which  z  is  negative  ;  '^b'^  —  v'^  is  to  be  taken  with  the 
positive  sign  for  a  point  on  the  half  of  the  biparted  hyperboloid  on  which  y  is 
positive,  and  with  the  negative  sign  for  a  point  on  the  half  on  which  y  is 
negative  ;  i^  is  to  be  taken  positive  for  a  point  on  the  half  of  the  biparted 
hyperboloid  on  which  x  is  positive,  and  negative  for  a  point  on  the  half  on 
which  X  is  negative,  and  that  the  remaining  six  are  to  be  always  positive.  It 
follows  that  our  Ellipsoidal  Coordinates  have  the  disadvantage  that  to  fully 
fix  a  point  Ave  need  to  know  not  merely  the  values  of  its  coordinates  A,  fx,  and 
V,  but  the  signs  of  sjc'^  —  /a^,  and  \Jb'^  —  iP-  as  well. 


252  ELLIPSOIDAL   HARMONICS.  [Art.  136. 

We  shall  see  later,  Art.  139,  when  we  come  to  introduce  what  we  may  call 
the  Normal  Ellipsoidal  Coordinates  a,  (3,  and  y  that  they  are  free  from  this 
disadvantage. 

It  is  to  be  observed  that  A  may  range  from  c  to  cc,  fi  from  b  to  c,  and  v  from 
-  b  to  b. 

The  element  of  length  perpendicular  to  the  Ellipsoid  is 


--l^=V&?s:^--  (^) 


The  element  of  Ellipsoidal  surface  is 


and  the  element  of  volume  is 

hi/iJh     \/(x:'  -  b')(x:'  -  c')(^'  -  b'')(c'  -  fi')(b'-v')(c-'-v') 

The   surface   integral  of  any  given  function  of  fi  and  v  taken  over  the 
ellipsoid  is 


ft  c 


+A(^,.)W  -  .')  V(^,  _  t,g  _  ;!g_  :^(,.  _  ,.)•  -^M,  (7) 

where  fi(lJ^,v),  /^(fJi',^),  f3(f^,v)  and  fi(fJi,v)  are  the  values  of  the  given  function 
on  the  four  quarters  of  the  ellipsoid  into  which  it  is  divided  by  the  planes  of 
(XY)  and  (XZ). 

Laplace's  Equation  proves  reducible  to 

(fi^ -  v^)Dl  V+  (X:' -  v')I)l  V  +  {X' -  ix^)D-'^  V=0  (8) 


Jdv  
^(b'-v^){c'-i^'')' 


(9) 


Chap.  VIII. ]  NORMAL   ELLIPSOIDAL   COORDINATES.  253 

a,  /3,  and  y  can  be  expressed  as  Elliptic  Integrals  of  the  first  class  and  are 


1    " 


y  =  F(^,sm-^f);  (10) 

I  mnrl  -  I  =r  r; 1  mod  -  h 

(A' -a) 


whence     A  =  — tt^ ( mod    )  =  '' ( mod  - ) 

snui  —  a)\         c/         cnaV         c/ 


/^  =  :Tf^("^of^(l-^2)  j'    ^-^^-sny^mod-')  (11) 


b 
dn/3' 


(v.  Int.  Cal.  Arts.  179,  192,  and  196). 

137.     If  in  (8)  Art.  136  we  assume    V=  L.M.N  where  L  involves  a  only, 
M  involves  (3  only,  and  N  involves  y  only,  (8)  can  be  written 

fi-  -  v'  cPL       A-  -  v'  (P3f      A-  -  fx'  cPN_ 

L      da'^      M     dp''^       N      dy'~  ^^ 

(1)  is  too  complicated  to  be  broken  up  by  our  usual  method. 
If,  however,  we  let 

substitute  in  (1)  and  make  use  of  the  fact  that  the  result  must  be  identically 
zero,  we  find  that  the  coefficients  are  zero  for  all  values  of  k  except  A*  =  0  and 
k  =  2,  and  that  ao  =  —  ^o  =  ^o ;  and  a^  =  —  &2  =  ^2  • 

Therefore  (1)  can  be  broken  up  into  the  three  equations 

_=(ao  +  «,A-)L 
-^  =  («o  +  (hV')N. 


254 


ELLIPSOIDAL    HARMONICS. 


[Art.  137. 


We  shall  find  it  convenient  to  take  ^2  ''^s  m.(in  -\- 1)  and  a^  as  —  {U^  +  •'-^)i^ 
whence  ' 


—  -  [m(m  +  1)A=^  -  (b'  +  c')p-]L  =  0 


dm 


af^2  +  ["K"^  + 1)/^'  -  (^^'  +  ''')pW=  0  h 


-  [m(m  +  l)v'  -  (b'  +  c')p]N=  0. 


(2) 


If  now  in  (2)  we  replace  a,  /3,  and  y  by  their  values  in  terms  of  X,  fi,  and 
V,  we  get 

(X^  -  b^-)  (X^  -c')~  +  X(X^  -b'-  +  \'-  c^)  '^^ 


dX' 


d^M 


^^^-l^X^,^-c-)—  +  ^.^^.--b-^Jr^^'-c^)■^^^ 


d\ 

[m(m  +  1)X-  -  (b'  +  c')p^L  =  0 
dM 


-  [m(m  +  l)fi'  -  {P  +  c')p^3I=  0 


(,._^,>)(,2_,.)^+,(,._^,.  +  ,2_,.)f^^ 


c?y 


(3) 


—  \_m{m  +  1)1^-  —  (&2  +  t'2)|)]iV^=  0.  ^ 

Whence  if  L  =  -E^,(X),  it  follows  that  3I=Ell(fi)  and  N=Ei;^{v),  and  that 

V^E^^(X)E^^{fi)Er.iv)  (4) 

is  a  solution  of  Laplace's  Equation,  (8)  Art.  136. 
The  equation 

(x^-  _  i;^) (x^  -  cO  ^,  +  a-(a-^  -  ^.^  +  a-'^  -  e=^)  -£ 


dx" 


[^m{m  +  l).r-'  —  {b""  +  c>]s;  ==  0         (5) 


is  known  as  Lame's  Equation,  and  Et;,(x)  as  a  Lame's  Fiinction  or  an  Ellij)- 
soidal  Harmonic.     We  shall  suppose  m  a  positive  integer. 

To  get  a  particular  solution  of  (5)  let  z  =  ^a^x^.     Substitute  in  (5)  and 
reduce  and  we  get 

\_k{k  +  1)  -  m{m  +  l)]a,  -  (5^  +  c^)  i{k  +  2)^  -py,^, 

+  h'e^ik  +  3)  (A-  +  4)a,^4  =  0.         (6) 

We  have  now  only  to  choose  a  sequence  of  coefficients  satisfying  (6),  and  we 
may  take  any  two  consecutive  coefficients  arbitrarily. 


Chap.  VIII.]  LAMe's   FUNCTIONS.  255 

(6)  which  is  ordinarily  a  relation  connecting  three  consecutive  coefficients 
reduces  to  a  relation  between  two  when  /.■  =  m.,  when  Ic  =  —  3,  and  when 
A'  =  —  4.  If  we  take  «,„  +  2  =  0j '^',h  +  4>  ^'m  +  «>  &c.,  will  vanish.  Let  a„^=^l. 
If  m  is  even  the  coefficient  of  ciq  in  (6)  will  be  zero  ;  if  p  has  such  a  value 
that  a_o  is  zero,  a_4,  a_6,  &c.,  will  be  zero,  and  there  will  be  no  terms  in 
the  solution  involving  negative  powers  of  x. 

If  we  write  the  values  of  a„,_2,  «m_4,  &c.,  by  the  aid  of  (6)  we  see  that 
a„,_2  is  of  the  first  degree  in p,  «„,_4  of  the  second  degree  in  j),  &c.,  and  «_2 

of  the  degree  tt  +  1  iii  i>-     There  are  then  77  +  1  values  of  p  which  we  shall 

call  ^1,  ^2  5^*3  5  &c.,  for  which  a_^  will  vanish,  and  for  which  our  solutions  will 
be  of  the  form 

:E^,(x)  =  a-'«  +  a,,^  _  2  X'"  -  -  +  (i,,^  _  4  X'"  -  ^  ^ \- <^ 

if  m  is  even. 

If  m  is  odd,  the  coefficient  of  «i  in  (6)  will  vanish  and  we  can  choose  j^  so 
that  a_i  shall  be  zero,  and  then  all  coefficients  of  lower  order  will  vanish. 

a_i  is  of  the  degree  — - —  in  p,  and  there  will  be  —^ —  values  2h,  i'25  2h} 
&c.,  of  2^  for  which 

^^^(x)  =  x"'  +  '(„,^,x"'-'  +  a^„_,x"'-'  H h  ^'i^:-. 

Following  Heine  we  shall  call  the  solution  just  obtained  Kl,',(x)  so  that 

KP(x)  =  X'"  +  c(,„  _  2  X'"  -  -  +  r/„,  _ ,  X"'  -  ^  H (7) 


terminating  with  Oq  if  vi  is  even,  and  with  ctiX  if  m  is  odd.  If  m  is  even, 
there  are  — +1  of  these  functions  K,>,['(x),  K^^-^(x),  &c.,  and  there  are  — ^~— 
of  them  if  m  is  odd.     The  coefficients  can  be  computed  by  the  aid  of  (6). 


If  in  Lame's  Equation  (5)  we  let  z  =  r\x-  —  U-  v/e  get  the  equation 

-  [(m  +  2) (m  -  l)x'  +  6--  -  (^^'  +  c'>]  V  =  0.         (8) 
Letting  v  =  ^a^x*  we  obtain  the  relation 

[k(k  +  3)  -  (m  +  2){m  -  l)-]a,  -  {(F-  +  c'Xik  +  2)^'  -7.]  +  c^(2/c  +  5)}a,^2 

+  ^-V(A-  +  3)(A-  +  4)«,^,  =  0.        (9) 


256  ELLIPSOIDAL   HARMONICS.  [Art.  137, 


Proceeding  exactly  as  before,  we  find  that  there  are  —  values  q^,  ?2>  ^S)  &c., 

of  j;  for  which    r  =  ./•"'-' +  ^?,„_ 3a,-'"-"  H V  ^xX    if    m    is  even,  and   — 7, — 

values  for  which  r  —  .r'"-'  +  a„,_^,x'"-^  H \- a^  if  m  is  odd. 


Calling  v\ix-—lr  L\n{x)  so  that 


ip(^)  =..  Vx--^'-[.r'"-'  +  '',„_, .T'"-"  +  «,„_5X"'-^  +  •  •  •],  (10) 

terminating  with  a-^x  if  m  is  even  and  with  a^  if  ?;i  is  odd,  we  have  — 
values  of  JS'/,;(a;),  namely  i,?,'(.x),  -^„?(a-'))  &c.,  of  the  form  (10)  if  m  is  even 
and  — ^— -  values  if  m  is  odd. 

By  interchanging  h  and  c  in  (8),  (9),  and  (10)  we  may  show  that  if 


Mlix)  =  s/x-  —  c-  [x"' - '  +  a,,^ _ 3 X'"  - •■=  +  a„, _ , x'" -' -\ ]  (11) 

there  are  —  values  of  ^,^(.r),  namely  J/,;;'(.r),  ^1,'^X^)^  ^^m(^')^  &c.,  of  the  form 

,   VI  +  1        ,         .  „         .        , , 
(11)  if  7)1  is  even  and  — r^ —   values  if  vi  is  odd. 


Finally  if  in  Lame's  Equation  (5)  we  let  z  =  r\J{x-  —  lr){x-  —  c-)  we  get 

{X-  -  h-)  (X--  -  C-)  ^.  +  3.r(x'2  -  U'  +  .T-  -  C-)  ^ 

-  \_{vi  +  3)(m  -  2),/^  -  (1/-  +  6'^)(/j  -  1)]^'  =  0.       (12) 
If  now  we  let  v  —  '±o,,x''  we  obtain  the  relation 
\_k{k  +  5)  -  {in  -  2)(m  +  3)Jr?, 

-  (//-'  +  r^)[(;.-  +  2)(/.-  +  4)  +  1  -pyi,^,  +  /A;^'(A-  +  3)(^-  +  4)a,,,  =  0.  (13) 

Proceeding  as  before  we  find  that  there  are  —  values  Si,   5.3,   s^,  &c.,   of  j^ 

for  which    ?>  =  ji-'"-- +  r/„,_4.)""-'' +  r/,„_ya:"'-''' H \- n^    if    ?«    is    even,   and 

^"  values  for  which  v  =  .r"'--  -)-  c?,„_^a-'"~^  +  •  ■  •  +  '■'i-'''  if  ^n  is  odd. 


Calling  rV(^— //-)(.r-  — (•-)  .X^;(.r)  so  that 


X^lx)  =  ^J{x'  -  1>'){J-  -  (")[.r"--  +  a„,_,x'"-'  +  a„_«.r"'-'^  +  •  •  •]        (14) 

terminating  with  Qq  if  7)i  is  even  and  with  UiX  if  ?«  is  odd,  we  have  —  values 
of  E;;,(x),  namely  -A^,';(a;),  ^''^(x),  JSr^f(x),  &c.,  of  the  form  (14)  if  w  is  even  and 
— - —  values  if  ?m  is  odd. 


Chap.  VIII.J 


TABLES    OF    ELLIPSOIDAL    HARMONICS. 


257 


Summing  up  our  results  we  see  that  there  are  2?«  +  l  Ellipsoidal  Harmonics 
E^^(x)  each  of  which  is  a  finite  sum  of  the  mth  degree  in  x,  or  in  x  and  \Jx^—U\ 
or  in  X  and  sjx^  —  c^,  or  in  x  and  '^x^  —  b^  and  >Jx'^  —  c\ 

It  was  proved  by  Lame  that  the  2m  +  1  values  of  j),  namely  ^^i,  ^j.,,  jj^,  &c., 
S'l)  ^2?  ^sj  <^c.,  ri,  rj,  r^,  &g.,  Si,  s.^,  s^,  &c.,  were  all  real,  and  by  Liouville  that 
they  were  all  different. 

We  give  tables  of  the  Ellipsoidal  Harmonics  for  m  =  0,  m  =  1,  m  =  2,  and 
m  =  3.  The  coefficients  were  obtained  by  the  aid  of  formulas  (6),  (9), 
and  (13). 

Eo(x)  E,(x) 


A(.r)  =:0 
Mo(x)  =  0 
N,(x)=0 


K,(x)=.r 

Mi(x)  —  yjx-  —  c- 
iVi(.r)  =  0 


E,(x) 

Kp(x)  =  x'-  -  ^  [fi'  +  r-  -  v/(/r'  +  c'Y 

-  3//-V,-] 

KS^(x)  =  x'  -  ^  [//^  +  c"'  +  n/(//-'  +  c^r 

-  SO-c'] 

L,(x)    =x\/x'  —  b'' 

M„(Jc)    =xsjx-  —  c- 

iY,(x)    =s/(a.^-i2)(,r-'_c^) 

A\(x) 


iq^(x) : 

-blx^- 

'•-)  -  n/4( 

''1 

c^)] 

^r'  +  fr)^'-15Z'-< 

^/-  +  cy  —  15F-1 

-  V'(^>-^  +  2cy  - 

-W 

Lp(x)   -. 

=  V1'^ 

—  b%x'-  — 

i(^^  +  2c^ 

+  \/(b'-\-2cy- 

-  5b- 

.•^)] 

3Ip(x)  -- 

=  Vx"^ 

-cXx-- 

1(2^-2  +  .^ 

-  sj{2b-'  +  cy  - 

-  5// 

.^)] 

Mp(x)  -- 

=  Va^ 

-c^[x^- 

1(2^-^  +  ^'^ 

+  V(2^»-^  +  c-T- 

-5^- 

r')] 

N,(x)    -- 

=W(: 

r--b')(x' 

-'>') 

258  ELLIPSOIDAL   HARMONICS.  [Art.  138. 

It  is  to  be  noted  that  since  in  the  solution  (4)  of  Laplace's  Equation, 

V=  EP(\)  El\{fi)  El>(v), 

we  have  the  same  m  and^j*  in  each  of  the  three  factors,  we  shall  have  to  deal 
merely  Avith  products  made  up  of  factors  of  the  same  form,  for  example, 

A'„f ^ (A)  ^„f ^-(m)  k:'x^)  ,  l:\x)  i,;^(/x)  l:;^x^) ,  &c. ; 

and  that  in  a  solution  of  the  form 

we  shall  have  for  a  given  m  just  2?«,  +  1  terms. 

138.     From   the  particular   solution  of   Lame's  Equation   [(5)   Art.   137] 
z  =  EfJx),  we  can  get  by  formula  (5),  Art.  18,  the  general  solution. 

dx 


/(lor 


Making  A^O  and  B  =  2m  -\-l  we  get  a  second  form  of  particular  solution  of 
Lame's  Equation,  z  —  Fl',(x)  where 

Fj;Jx)  =  (2m  +  l)EZ(x)  (  ,—  '  = (2) 

We  shall  call  Fp(x)  a  Lame's  Function  of  the  second  kind. 

It  is  easily  seen  to  approach  the  value  zero  as  x  is  indefinitely  increased. 

EXAMPLES. 

1.    If  an  ellipsoidal  conductor  is  charged  with  electricity,  and  is  found  to 
be  at  potential  Vq,  show  that  since  Fo=  FoA'o(A), 

F=  V,K,(\)K,{p)K,{v)  =  Vo 

.  at  an  internal  point,  and 

dx 


V=  l\K,{^.)K,(y)^K,(X)J^ 


sJ(x^-bXx'-c^)lK,{x)f 


=  V, 


{^{x'-h'^){x^-c')lK,(x)Y-\ 

r  r dx ._  r (ix n  _        \c a/ 

LJ  N/(x2-/A)(.r--^-r2)   •  J  N/(.r2-/.^)7^2"^:^J  ~     \(^_^  sin-^  -)  ' 
"  \c  xJ 


Chap.  VIIL]  NORMAL    ELLIPSOIDAL    COORDINATES.  259 


whence 


V^  V, 


V.  (10)  Art.  136. 


2.    Find  the  value  of  the  potential  function  at  an  external  point  due  to  the 
attraction  of  a  solid  homogeneous  ellipsoid  (v.  Art.  135). 
Observe  that 

f  /.2     I      -.2  o  72         —I 


and  that 


where  31  is  the  mass  of  the  ellipsoid. 
dx 


Ans.  V=M 


iJ^W^ 


V/(x-2  — Z*2)(a;2  — c2) 


,  ^  — rA7-(At)A>(^)A>(A)  f-=_S= 

2n/(A-  +  c2)2  —  3^*VL  "  '  '  y  \J(x-  —  //-) (.r^  —  c').(iq\x)y 

^        ^  '    ^  Y  v'(^2-Z-^)(x=^-c2).(A>(x-))0  i 
139.     If  for  the  sake  of  brevity  we  represent  -  by  k,  and  (1 rj    by 


k'  in 


the  formulas  (11)  Art.  136  we  have 

^  =  ^'^^^^^°^^^^'     ^  =  ch./3(Ldk'y     -  =  ^'«"yOnodA-)  (1) 

and  from  these  we  get  without  difficulty  (v.  Int.  Cal.  Art.  192) 


v^a"^^^7;^  = 


ck' 


en  a  (mod  A-)  dn/?    ^  '^' 


\/b-  —  v-^^^b  Q,\iy  (mod  A-:),  VA'  —  c-  ^  -^^ (mod  k), 

sjc-  —  u-  =  -- — -^  (mod  A-'),       V^'"  —  V-  =  c  dn  y  (mod  /c). 
dn  y8  ^  V  / 


(2) 


260  ELLIPSOIDAL   HARMONICS.  [Art.  lit). 

If  Ave  let  a  range  from  0  to  A',  and  /3  from  0  to  2/v ',  and  y  from  0  to  4A', 

where  7i  and  7v' are  the  complete  Elliptic  Integrals  F{k,  —  \  and  i^(A-'.  — ) 

respectively,  (a,  /3,  y)  may  represent  any  point  in  space,  and  there  Avill  be  no 
ambiguity  in  sign  (v.  Art.  136). 

We  may  note  that  if  0  <  /?  <  A',  z  is  positive  ;  if  K'  <.  /3  <.  -K',  z  is 
negative;  if  0  <  y  <  A^,  x  and  y  are  both  positive;  if  Jr<y<2A,  x  is 
positive  and  y  negative  ;  if  2 A'  <  y  <  SA',  x  and  y  are  both  negative  ;  and  if 
37ir<  y  <  4  A,  a;  is  negative  and  y  positive  (v.  Art.  136). 

We  can  write  the  values  in  (4),  (5),  (6),  and  (7),  Art.  136,  more  neatly  by 
bringing  in  a,  /3,  and  y.     We  get 

1 


dn  =  -  V  ( A-  —  ix-)  (A-  —  V-)  da,  (3) 


dS  -  -,  (yu^  -  z.2)n/(A^'  -  ix')  (A^  -  ^^)  dlidy,  (4) 

fZ^.  =  -  (A2  —  ^2)  (A-  —  V-)  (jjC-  —  V-)  dadfidy.  (5) 

For  the  integral  of  any  function  of  a,  /3,  and  y  over  the  ellipsoid  a  =  a,,,  we 
shall  have 

■2K'  4A- 

jF(a,/3,y)dS  =  ^^Jd/3jF(a„(3,y){fi'  -  ^)\/(X' -  fji'){X' -  v^)dy.         (6) 

140.     If  we  make  use  of  the  formula  (2)  Art.  92 

Ji  UD„  V  -  VD„  U)dS  =  0  (1) 

and  take  as  our  closed  surface  any  given  ellipsoid,  we  can  get  a  very  important 
result. 

If  U=Ei',(X)E^l(/x)F;;(v)     and      V  =  E,1(\)  E;i(/x)  F^  (v) 

then  v-r=v-r=o. 


D„  U=  D^  UD^a  =  F,';,(fi)Fi:,(v) 


and  X»„  V=  A  VI)„a  =  F,'i  (fi)  E^  (v) 


dEP{\) 


da     yJ(^x''  —  fi-)(X'  —  v') 

dE:>{\)  c 


da       v/(A2_^2)(;^2_j,2^ 

UD„V—VD„U 

=  E/;(f.)E^,(v)E,^(fx)E:i(v)(^El^,(X)  '^^ -  E:!(X)  '^^) 


da       /S/(A2  —  yLl2)(X^'_i;2) 


Chap.  VIII.]  DEVELOPMENT    IN    LAME  S    FUNCTIONS.  261 

Integrating  UD„V  —  FI>„?7over  the  whole  ellipsoid,  and  writing  the  result 
equal  to  zero,  we  have 

2A"  4  A' 


2A"  4A- 

Hence 


unless  ES(K)^-Ji;(>.)'^  =  0.  (3) 

But  as  our  ellipsoid  may  be  taken  at  pleasure,  A  and  a  are  unrestricted,  and 
if  (3)  is  true  it  must  be  trvie  identically. 
If  we  divide  (3)  by  [Ep(\)J  it  becomes 

drE,ux)-\     ,^       ^    E,ia) 

;^L:^-I^  ;^  =  a  constant; 

and  this  obviously  cannot  be  true  unless  w  =  )ii  and  7  ^=p. 

EXAMPLES. 

1.  Show  that  it  follows  from  (2)  Art.  140  that 

A"  A- 

Jd(3  CEP(fji)EP(v)E,'l(lji)E,?(v)(^'  -  v-)dy  =  0. 

-K         -K 

Suggestio7i  : 

2K'  K' 

jE„P,(fi)E,^(fi)(fx'  -v')d(3  =jEl,](fi)E:i(,x)(fi^  -  v'-)dS 

•IK' 

+  fEi:,(fi)E;!(fi)(f,'-  -  i,^-)dB. 

A'' 

If  in  the  last  integral  we  replace  /3  by  /3  +  2 A''  it  becomes 

n 
-A'' 

V.  Arts.  136  and  139  and  Int.  Cal.  Art.  196. 

2.  Show  that 

2A"  4  A'  A"  A- 

Jdl3f[ES,(fi)Ej:,(v)J(ti'  -  v')dy  =  8jd(3j[Ei:(fi)Ei;,(v)]\fi'  -  v^ly. 


262  ELLIPSOIDAL   HARMONICS.  [Airr.  141. 

141.  We  can  now  solve  the  problem  of  finding  the  value  of  Tat  any  point 
in  space  when  it  is  given  at  all  the  points  on  the  surface  of  the  ellipsoid 

We  have  first  to  develop  in  Ellipsoidal  Harmonics  a  function  of  yu.  and  v  or 
rather  of  a  and  /3  given  at  all  points  on  the  surface  of  the  ellipsoid  in  question ; 
and  this  is  now  easily  accomplished  by  our  usual  method,  which  leads  us  to 
the  result 

m=x  A-  =  2m  +  1 


where  J,,,^.  = " ^^^ (2) 


Our  final  solution  is 


m  =  »  A-  =  jm  +  1  p 


at  an  internal  point; 


at  an  external  point. 

Lame   has  proved  rather  ingeniously  that 


p^j'[E':!^Xf^)E!:'(^)j(fi'  -  vyiy 


can  always  be  found  and  that  it  is  equal  to  —  multiplied  by  a  rational  integral 

function  of  the  coefficients  of  ^i^'(a')  and  of  c-  and  (-)  • 

Of  course  the  labor  of  obtaining  even  a  few  terms  of  the  development  of  a 
function  that  is  in  the  least  complicated  is  enormous. 

142.     If  in  Laplace's  Equation  (8)  Art.  136  we  let    V=Ep(\)U  supposing 
?7  to  be  a  function  of   /3  and  y  only,  we  get  after  replacing  — '"!, 

by  its  value  m,(m  +  1)A-  —  (b'  +  c->  [v.  (2)  Art.  137] 

(X'  -  v'-)Dl  U+(\'-  fJ?)D';  U  +  (fjr  -  v') [m(m  +  1)A—  (Ir  +  c')p']  T  =  U  ;      (1) 


Chap.  VIIL]  CONICAL   COORDINATES.  263 

and  since  by  hypothesis    U  is  independent  of  A,  the  coefficient  of  X^  in  (1) 
must  vanish.     Hence 

Dl U  +D';U^  (fx'  -  v') m(m  -]-l)U=0.  '  (2) 

Of  course   U=  UP(fi)EP(v)  will  satisfy  (2). 

EXAMPLES. 

1.  Substitute  U=  EP(fi)EP(v)  in  (2)  Art.  142  and  by  the  aid  of  (2)  Art.  137 
show  that  the  equation  (2)  Art.  142  is  satisfied. 

2.  Obtain  (2)  Art.  140  directly  from  (2)  Art.  142. 

.3.    Co7iical  Coordinates.     Consider  the  system  of  coordinates  defined  by  the 
equations 


fx-        jx-  —  Ir        fx-  —  r- 


(1) 


V-       v~  —  h~       V-  —  c- 
diere   r^  >  /x-  >  Ir  >  ir. 
Show  that 

^ r^x-v'         2 ''"{y?  —  lr){v~  —  Jr)         , 't^iix"  —  (^^v-  —  c^)  . 

^h-  ,.2(^2  _  ^2)  '         Ih-  ^(^2_^2)  '       /'3-1- 

Laplace's  Equation  is 

Dl  V  +  Dl  V  +  (fx'-  -  v')D,.(>^I),.  F)  =  0  (2) 

=   f-=J^==     and      fi=  t  ^'^ 

J  \/(fx'  -  b')  (c'  -  fx')  J  \/(f,^  -  v')  (c^  -  z.^') 

If   F=  U.B  (2)  breaks  up  into 


where 


(^^'^)=.n.(m,  +  l)E,  (3) 


d_/  „dE 
dr 


Dl U -f  D'^  U  +  m{m  +  1) {fx'  -v-)U=  0.  (4) 

(3)  gives  B  =  Ar'"  +  Br-"'-\ 

(4)  gives  U=EP(/x)EP(v)  (v.  Art.  142). 
So  that  a  solution  of  (2)  is 

r=  Ar"'EP(fx)El](v). 

But  since  (2)  is  Laplace's  Equation,  ir=^r'«F^(/x,  ^),  if  expressed  in 
Conical  Coordinates,  must  satisfy  it,  consequently  EP(fx)EP(v)  must  be  simply 
a  Spherical  Harmonic  of  the  ?/ith  degree. 


264  ELLIPSOIDAL    HARMONICS.  [Art.  143. 

Toroidal   Coordinates. 

143.     Any  pair  of  circles  belonging  to  the  orthogonal  system  obtained  and 
figured  in  Art.  46  can  be  represented  by  the  equations 

2ax         x~  +  ?/-  +  «- 


sinh  a  cosh  a 

„  ,    .,      A  (1) 

2ft//  ^3-  +  //-  — 
sin  /3  cos  /3 

if  we  take  2a  instead  of  2  as  the  distance  between  the  points  common  to  the 
second  set  of  circles. 

If  we  rotate  the  system  about  the  axis  of  i/  we  get  a  set  of  spheres  and  a 
set  of  anchor  rings  which  cut  orthogonally.  These  and  a  set  of  planes  through 
the  axis  of  revolution  will  form  an  orthogonal  system  of  surfaces,  and  the 
parameters  corresponding  to  them  may  be  taken  as  a  set  of  curvilinear 
coordinates  and  may  be  called  Toroidal  Coordinates. 

If  we  take  the  axis  of  the  system  as  the  axis  of  Z,  the  equations  of  a  set  of 
the  surfaces  may  be  written 

4ft-(a,--  +  //-)  ^  r-^"  +  !/  +  -^-  +  'rj 
sinh-  a  cosh-  a 

2az        X-  +  //■  -\-  z-  —  <r 


(2) 


sin  /3  cos  y8 

y  =  x  tan  y  J 

a,  13,  and  y  being  regarded  as  the  coordinates  of  a  point  of  intersection  of  the 
three  surfaces. 

Finding  Laplace's  Equation  in  the  usual  manner  we  get 

_  a  sinh  a  cos  y  _  a  sinh  a  sin  y  a  sin  ^ 

'^~  cosh  a  :p  cos^'  '^  ~  cosh  a  qi  cos /3 '  "       cosharpcosyS 

sinh  a  ,        ,     ^  « cosh  a 


r  =  six-  +  ir  = \ ;,'  a-\-  z  ctn  i^ 

^     ^  -^       cosh  a  If  cos  13  ^      cosh  a  ^:  cos  13 

cosh  a  q:  cos  ^  ,        cosh  a  q:  cos  (3  _  cosh  a  zp  cos  13 . 

^  a  '  a  a  smh  a 

and  Laplace's  Equation  becomes 

r       a  sinh  a        -r,  t-"1    i    n  f       ^'  ^^^^^^  "        r.   t^~1 


Chap.  VIIL]  TOROIDAL   HARMONICS.  265 

D„(rD„  V)  +  D,(rD,  F)  +  -^  »-i>;  V=0.  (2) 

We  cannot  proceed  further  by  our  usual  method,  for  the  assumption  that  V 
is  a  function  of  a  alone,  or  that  V  is  a  function  of  ft  alone,  proves  to  be 
inadmissible.  Indeed,  not  only  are  a,  ft,  and  y  not  thermometric  parameters 
(v.  Art.  133),  but  no  thermometric  parameters  exist,  and  no  possible  distribu- 
tion can  make  our  anchor  rings  or  our  spheres  a  set  of  equipotential  surfaces. 

We  can,  however,  simplify  (2).     It  can  be  written 

i>a(  vs/;-)  +  i)i(  FN/;) + —^  D;(  FN/;)  -  i^^aV; + div;)  -  o.     (3) 

D'^^ r -\-  iJ'^s/ )'  proves  equal  to  —       .  j    hence  if   U ^=  V^r  (3)  becomes 

sinh^  a(DlU+  Z»|  U)  +  J)^  U+\U=  0,  (4) 

for  which  particular  solutions  can  readily  be  found  by  our  usual  process. 
(4)  can  be  broken  up  into  the  three  equations 

^'+(/M  +  i)^V=0  (5) 

dy- 

sinh^ a  ——  —  [m(m  +  1)  +  ir  sinh-  alL  =0.  (7) 

da-       '-     ^ 

X:=  A  cos(m  -\-  4-)y  +  -^  sin(?/^  +  4-)y 

31=  Jj  cos  nft  +  Bi  sin  7ift. 

If  we  introduce  into  (7)  x  =  ctnh  a  it  becomes 

a  solution  of  which  is 

L  =  piix)  =  (1  -  x^y  "^"i  (^-  ^^'^^  1*^2). 

It  is  to  be  noted  that  since  ctnh  a  is  greater  than  1 

T,  ,     X        .-       .       cZ"P™('ctnha) 

P"(ctnh  a)  =  i2  csch"  a     ,,"*:, — r-^- 
'"^  ^  (c?  ctnh  a)" 


266  ELLIPSOIDAL    HARMONICS. 

The  constant  coefficient  i-  can  be  rejected  and  we  get 

U=  \_A  cos(vi -\-h)y-\-S  sin(7?i  +  h)y](Ai cos nji-\- B^  sin  ?i/3)csch" 


(/"P,„(ctnha) 
((^ctnha)" 
as  a  particular  solution  of  (4). 


1  T.  .  .  1     N  1  .    rf"P„Yctnha) 

-  P,  (ctnh  a)  =  csch"  a  — --^^^^ — -^ 
.'-'      ^  (fZctnha)" 


I 
has  been  called  a  Toroidal  Harmonic. 

EXAMPLES. 

1.  Given  the  value  of  the  potential  function  at  all  points  on  the  surface  of 
an  anchor  ring ;  find  its  value  at  any  point  within  the  ring. 

Suggestion:    If  V=f((S,  y)  when  a^ao<  the  function  to  be  developed  is 

and  the  development  will  be  in  a  double  Fourier's  Series  (v.  Art.  71). 

2.  Show  that  if  we  let  a  range  from  0  to  oc,  ^  from  —  tt  to  tt,  and  y  from 
0  to  27r,  each  of  the  double  signs  on  page  264  may  be  replaced  by  the  minus 
sign  without  loss  of  generality. 


CHAPTER   TX.* 


HISTORICAL     SUMMARY. 


The  method  of  development  in  series  which  has  enabled  ns  in  the  preceding 
chapters  to  solve  problems  in  various  branches  of  mathematical  physics,  had 
its  origin,  as  might  have  been  expected,  in  the  theory  of  the  musical  vibrations 
of  a  stretched  string.  It  was  in  the  year  1753^  that  Daniel  Bernoulli 
enunciated  the  principle  of  the  coexistence  of  small  oscillations,  which,  in 
connection  with  Taylor's  and  John  Bernoulli's  theory  of  the  vibrating  string, 
led  him  to  believe  that  the  general  solution  of  this  problem  could  be  put  in 
the  form  of  a  trigonometric  series.  This  principle  also  led  him  and  Euler  to 
treat  in  a  similar  manner  the  problems  of  the  vibration  of  a  column  of  air  and 
of  an  elastic  rod.  The  problem  of  the  vibration  of  a  heavy  string  suspended 
from  one  end  was  also  treated  in  the  same  manner  by  these  mathematicians 
and  deserves  special  mention  here  as  in  it  Bessel's  functions  of  the  zeroth 
order  appear  for  the  first  time.^  In  none  of  these  cases,  however,  was  any 
method  given  for  determining  the  coefficients  of  the  series. 

This  last  remark  also  applies  to  the  more  complicated  problems  of  the 
vibration  of  rectangular  and  circular  membranes,  which  were  discussed  by 
Euler  ^  in  1764,  and  in  the  last  of  which  the  general  Bessel's  functions  of 
integral  orders  occur. 

It  is  in  problems  connected  with  astronomy  that  the  first  completely 
successful  application  of  the  method  here  considered  occurs.  Legendre  in  a 
paper  published  in  the  Memoires  des  Savants  Strangers  for  1785,  first 
introduced  the  zonal  harmonics  P^  and  applied  them  to  the  determination  of 
the  attraction  of  solids  of  revolution.  He  was  followed  by  Laplace,  who  in 
one  of  the  most  remarkable  memoirs  ever  written*  determined  the  potential 
of  a  solid  differing  but  little  from  a  sphere  by  means  of  the  development 
according  to  the  spherical  harmonics  y,„. 

1  See  two  articles  by  Bernoulli  and  one  by  Euler  in  the  Memoirs  of  the  Academy  of 
Berlin  for  this  year. 

2  See  the  Transactions  of  the  Academy  of  St.  Petersburg  for  1732-33,  1734  and  1781. 

3  Transactions  of  the  Academy  of  St.  Petersburg. 

4"Th^orie  des  attractions  des  sph^roides  et  de  la  figure  des  Plan  fetes"  Mfemoii'es  de 
I'acadfemie  des  sciences  1782.  This  article,  although  bearing  an  earlier  date  than  that  of 
Legendre,  was  really  inspired  by  it.  It  is  here  that  "Laplace's  equation"  first  appears, 
occurring,  however,  only  in  polar  coordinates. 

*  See  preface. 


268  HISTORICAL    SUMMARY. 

Very  closely  related  to  this  problem  is  Gauss's  celebrated  treatment  of  the 
theory  of  terrestrial  magnetism/  which  we  will  for  that  reason  mention  here, 
although  it  was  not  published  until  more  than  half  a  century  later.  This 
paper  is  particularly  noteworthy  as  it  contains  a  numerical  application  of  the 
method  on  a  larger  scale  than  has  ever  been  attempted  before  or  since. 

After  the  researches  of  Legendre  and  Laplace  there  was  a  pause  of  a 
quarter  of  a  century  until  in  1812  Fourier's  extensive  memoir  :  Theorie  du 
mouvement  de  la  chaleur  dans  les  corps  solides  was  crowned  by  the  French 
Academy.  Although  not  printed  until  the  years  1824-26,^  the  manuscript  of 
this  work  was  in  the  meantime  accessible  to  the  other  French  mathematicians 
presently  to  be  mentioned.  The  first  part  of  this  memoir,  which  was  repro- 
duced with  but  few  alterations  in  the  Theorie  analytique  de  la  chaleur  (1822), 
contains  a  treatment  of  the  following  problems  and  of  practically  all  of  their 
special  cases  : 

(a)  The  one  dimensional  flow  of  heat.  (Ij)  The  two  dimensional  flow  of 
heat  in  a  rectangle,  (c)  The  three  dimensional  flow  of  heat  in  a  rectangular 
parallelopiped.  (d)  The  flow  of  heat  in  a  sphere  when  the  temperature 
depends  only  on  the  distance  from  the  centre,  (e)  The  flow  of  heat  in  a 
right  circular  cylinder  when  the  temperature  depends  only  on  the  distance 
from  the  axis.  In  these  problems  not  merely  the  simpler  boundary  conditions 
are  considered  but  also  the  question  of  radiation  into  an  atmosphere.  In 
special  cases  of  the  first  three  problems  just  mentioned  (when  one  or  more 
dimensions  become  infinite)  the  series  degenerate  into  "  Fourier's  integrals." 

More  important  even  than  any  of  these  special  problems  is  the  great 
advance  which  Fourier  caused  the  theory  of  trigonometric  series  to  make. 
In  a  posthumous  paper  Euler  had  given  the  formulae  for  determining  the 
coefficients,^  but  Fourier  was  the  first  to  assert  and  to  attempt  to  prove  that 
any  function,  even  though  for  different  values  of  the  argument  it  is  expressed 
by  different  analytical  formulae,  can  be  developed  in  such  a  series.  The  fact 
that  the  real  importance  of  trigonometric  series  was  thus  for  the  first  time 
shown  justifies  us  in  associating  Fourier's  name  with  them,  although,  as  we 
have  seen,  they  were  known  long  before  his  day. 

Fourier's  results  were  extended  by  Laplace  in  1820*  to  the  general  (unsym- 
metrical)  case  of  the  flow  of  heat  in  a  sphere,  and  by  Poisson*  (1821)  to  the 
unsymmetrical  flow  of  heat  in  a  cylinder. 

1  Resultate  aus  den  Beobachtungen  des  magnetischen  Vereins  im  Jahre  1838.  Leipzig, 
1839.     Reprinted  in  Gauss's  collected  works,  Vol.  V.,  p.  121. 

2  M^moires  de  racaddmie  des  sciences  for  1819-20  and  1821-22. 

3  Lagrange  had  practically  determined  these  coefficients  long  before  but  failed  to  notice 
what  he  had  got. 

4  Connaissance  des  Temps  pour  Tan  1823. 

5  Journal  de  I'ficole  Polytechnique,  19*^  Cahier.  Although  the  final  forms  to  which  Poisson 
reduces  his  results  are  similar  to  Fourier's,  his  methods  are  very  different. 


ELLIPSOIDAL    HARMONICS.  •  209 

In  1835  Green  published  a  paper  ^  in  which  the  method  we  are  considering 
is  employed  to  determine  the  potential  of  a  heterogeneous  ellipsoid.  This 
paper,  in  which  the  analysis  is  performed  at  once  for  space  of  n  dimensions, 
anticipates  much  that  was  subsequently  done  by  others,  but  has  failed  to 
exert  an  influence  proportional  to  its  importance. 

At  about  this  time  Lame  began  a  series  of  publications  which  have  con- 
nected his  name  inseparably  with  the  problem  of  the  permanent  state  of 
temperature  of  an  ellipsoid.  In  the  first  of  these  -  the  equation  V'^  F  =  0  is 
transformed  to  ellipsoidal  coordinates  and  is  then  broken  up  into  three 
ordinary  differential  equations.  The  rest  of  the  solution,  however,  is  hardly 
touched  upon.  Lame's  most  important  work  on  this  subject^  was  published 
in  Liouville's  Journal  in  1839,  and  in  it  the  complete  solution  of  the  problem 
is  given.  Lame  clearly  shows  in  this  paper  how  he  arrived  at  his  solution,  by 
considering  first  the  simpler  case  of  a  sphere  where,  instead  of  the  polar 
coordinates  6  and  </>,  the  parameters  of  two  families  of  confocal  cones  of  the 
second  degree  are  used  as  coordinates.  This  system  of  curvilinear  coordinates, 
which,  Avhen  applied  to  the  complete  sphere,  merely  gives  the  old  results  of 
Laplace  in  a  new  form,  is  barely  mentioned  in  Lame's  later  publications.  In 
the  same  volume  of  Liouville's  Journal  Lame  published  a  second  paper  in 
which  he  applies  his  results  to  the  special  cases  of  ellipsoids  of  revolution. 

These  two  papers  form  the  starting-point  for  a  series  of  articles  on  the 
same  subject  by  Heine  and  Liouville.  Heine  in  his  doctor  dissertation*  (1842) 
determined  the  potential  not  merely  for  the  interior  of  an  ellipsoid  of 
revolution  when  the  value  of  the  potential  is  given  on  the  surface,  but  also 
for  the  exterior  of  such  an  ellipsoid  and  for  the  shell  between  two  confocal 
ellipsoids  of  revolution.  Even  in  the  first  of  these  problems,  which  is 
equivalent  to  that  of  Lame,  he  simplified  Lame's  solution  materially  by 
showing  that  the  functions  used  may  be  reduced  to  spherical  harmonics, 
while  in  the  other  two  problems  he  introduced  spherical  harmonics  of  the 
second  kind,  which  were  then  new.     Shortly  afterwards^  Heine  and  Liouville 

1  "  On  the  determination  of  the  exterior  and  interior  attraction  of  ellipsoids  of  variable 
densities."     Transactions  of  the  Cambridge  Philosophical  Society. 

2  M^moires  des  Savants  Strangers,  Vol.  V.  Although  the  volume  is  dated  1838  this  paper 
(which  was  reprinted  in  Liouville's  Journal,  1837)  must  have  appeared  at  least  as  early  as  183-5. 

3  "Sur  I't^quilibre  des  Temperatures  dans  un  ellipsoide  k  trois  axes  in^gaux."  An  article 
by  the  same  author  on  the  two  dimensional  potential  will  be  found  in  Vol.  I.  of  this  Journal. 

4  Reprinted  in  Crelle's  Journal,  Vol.  26  (1843). 

In  the  same  Journal  for  1847  F.  Neumann  discussed  the  related  problem  of  the  magnet- 
isation of  a  soft  iron  ellipsoid  of  revolution. 

^  Heine:  Crelle's  Journal,  Vol.  29,  1845.  Liouville:  Liouville's  Journal,  Vol.  X., 
1845,  and  Vol.  XI.,  1846.  For  a  treatment  of  the  problem  of  the  potential  of  an  ellipsoidal 
shell  by  means  of  a  development  of  -  in  terms  of  Lamp's  functions,  see  a  paper  by  Heine 
in  Crelle's  Journal,  Vol.  42,  1851. 


270  HISTORICAL    SUMMARY. 

published  simultaneously  two  papers  in  which  they  arrived  independently  of 
each  other  at  about  the  same  results.  In  each  of  these  papers  attention  is 
called  to  the  fact  that  the  product  of  two  Lame's  functions  is  a  spherical 
harmonic,  and  this  fact  is  made  use  of  to  throw  Lame's  solution  of  the 
problem  of  the  permanent  state  of  temperatures  of  an  ellipsoid  into  a  more 
elementary  form.  Besides  this  the  second  solution  of  Lame's  equation  is 
introduced  for  the  sake  of  solving  the  potential  problem  for  the  exterior  of 
the  ellipsoid. 

In  thus  following  up  the  theory  of  heat  and  the  related  potential  problems, 
we  have  lost  sight  of  the  question  of  small  vibrations,  to  which  during  the 
early  part  of  the  century  a  great  deal  of  attention  had  been  devoted  by 
Poisson,  who  frequently  made  use  of  the  method  of  development  in  series. 
In  his  memoirs^  most  of  the  problems  left  unfinished  by  Bernoulli  and  Euler 
are  thoroughly  treated,  as  well  as  various  slight  modifications  of  them. 
When,  however,  he  attacked  the  problem  of  the  vibration  of  an  elastic  plate 
he  was  unable  to  make  much  progress,  owing  in  part  to  the  erroneous  form  of 
his  boundary  conditions.  He  was,  nevertheless,  able  to  solve  the  problem  of 
the  symmetrical  vibration  of  a  free  circular  plate.  The  complete  theory  of  the 
vibration  of  a  free  circular  plate  was  first  given  by  Kirchhoff.^ 

Passing  now  to  a  new  subject,  the  theory  of  the  equilibrium  of  an  elastic 
spherical  shell,  we  find  a  solution  by  Lame  in  Liouville's  Journal  for  1854, 
and  by  Sir  William  Thomson  (1862)  in  the  Philosophical  Transactions  for 
1863.  Both  of  these  papers  consist  of  an  application  of  the  spherical  harmonic 
analysis  to  this  rather  complicated  problem.  Thomson,  however,  considers 
besides  Lame's  problem  certain  related  questions  and  the  form  of  his  analysis 
is  very  different  from  Lame's,  being  of  the  same  nature  as  that  used  in  the 
Appendix  B  of  his  Natural  Philosophy  of  which  we  shall  have  to  speak 
presently.  These  investigations  form  the  starting  point  for  a  number  of 
recent  memoirs  among  which  those  of  G.  H.  Darwin  on  cosmographical 
questions  deserve  special  mention. 

Closely  related  to  this  last  mentioned  problem  is  the  theory  of  the  small 
vibrations  of  an  elastic  sphere.  While  the  simplest  case  of  this  problem  was 
treated  by  Poisson  in  the  memoir  referred  to  above,  the  general  solution  has 
been  only  recently  obtained  by  Jaerisch  (1879)^  and  Lamb  (1882).*  The 
functions  involved  are  the  same  as  those  which  occur  in  the  problem  of  the 
non-stationary  flow  of  heat  in  a  sphere  as  solved  by  Laplace. 

The  Appendix  B  of  Thomson  and  Tait's  Natural  Philosophy,  ^  to  which  we 
have  already  referred,  deserves  to  be  regarded  as  one  of  the  most  important 

1  See  especially  the  one  in  the  M^moires  de  I'acad^mie  des  sciences,  Vol.  VIII.,  1829. 

2  Crelle's  Journal,  Vol.  40,  1850.  ^  Crelle's  Journal,  Vol.  88. 

*  Proc.  Lond.  Math.  Soc.  ^  First  edition,   1867.      This  appendix 

was  evidently  written  as  early  as  1862,  as  Thomson  refers  to  it  in  the  memoir  quoted  above. 


TOKOIDAL   AND    CONAL    HARMONICS.  271 

contributions  to  the  general  theory.  The  way  in  which  spherical  harmonics 
are  introduced  (as  homogeneous  functions  of  the  rectangular  coordinates)  was 
then  new/  and  the  sokition  of  the  potential  problem  for  a  variety  of  new 
solids  was  indicated ;  viz.,  for  solids  whose  boundaries  consist  of  concentric 
spheres,  cones  of  revolution,  and  planes.  We  shall  have  more  to  say  presently 
concerning  the  method  employed  for  the  solution  of  these  problems. 

Although  connected  only  indirectly  with  the  theory  we  are  discussino-,  it 
will  be  well  to  mention  at  this  point  the  method  of  electrical  images  which  is 
also  due  to  Sir  William  Thomson  (1845).  This  method  enables  us  to  solve 
many  potential  problems  for  the  inverse  of  any  solid  when  once  we  have 
solved  it  for  the  solid  itself.  By  means  of  this  method  most  of  the  solutions 
of  potential  problems  obtained  by  our  method  may  be  applied  at  once  with 
very  little  modification  to  systems  of  curvilinear  coordinates  derived  by 
inversion  from  those  we  have  used.  It  will  not  be  necessary  to  mention 
separately  problems  of  this  sort,  as  it  is  clearly  immaterial  whether  they  be 
solved  directly  or  by  means  of  the  method  of  inversion.^ 

Eeturning  now  to  the  Continent,  we  find  as  the  next  important  question 
taken  up  the  problem  of  the  potential  of  an  anchor  ring.  The  first  publication 
on  this  subject  is  a  monograph  by  C.  Neumann  ^  (1864),  but  in  Riemann's 
posthumous  papers  which  were  not  published  until  1876,  ten  years  after  his 
death,  will  be  found  a  short  fragment  on  this  subject,  which  {cf.  the  last  page 
of  Hattendorf's  edition  of  Riemann's  lectures  :  '<  Partielle  Differentialglei- 
chungen  ")  would  appear  to  date  back  to  the  winter  1860-61.  This  fragment 
is  of  peculiar  interest,  as  the  opening  paragraphs  clearly  show  that  Riemann 
had  in  mind  an  extended  article  on  the  fundamental  principles  of  our  subject. 

We  will  next  mention  two  papers  by  Mehler  in  which  the  functions  known 
as  ^'  conal  harmonics,"  which  had  already  been  introduced  by  Thomson  in  the 
Appendix  B  above  mentioned,  were  applied  to  the  solution  of  two  problems  in 
electrostatics.  The  first  of  these  papers  ■*  (1868)  deals  with  the  solid  bounded 
by  two  intersecting  spheres,  while  in  the  second^  (1870)  the  infinite  cone  of 
revolution  is  treated.  Both  of  these  problems  are  essentially  different  from 
those  discussed  in  the  ''  Appendix  B,"  inasmuch  as  the  infinite  series  which 
we  usually  have  degenerate  in  these  cases  into  definite  integrals,  just  as  they 
do  in  some  simpler  cases  treated  by  Fourier.  The  later  of  the  two  papers 
just  quoted  also  contains  valuable  information  concerning  the  nature  of  the 

1  The  same  method  was  used  at  about  the  same  time  by  Clebsch. 

-  A  case  in  point  would  be  the  potential  problem  for  the  shell  between  two  non-intersectino- 
eccentric  spheres,  since  these  spheres  can  be  inverted  into  concentric  spheres.  This  problem 
was  treated  directly  by  C.  Neumann  in  a  monograph  published  in  Halle  in  1862. 

3  "  Theorie  der  Elektricitiits-  und  Warme-Vertheilung  in  einem  Ringe."     Halle.       ^ 

4  Crelle's  Journal,  Vol.  68,  1868. 

5  Jahresbericht  des  Gymnasiums  zu  Elbing. 


272  HISTORICAL    SUMMARY. 

solution  of  similar  problems  for  the  hyperboloids  and  paraboloids  of  revolu- 
tion.    The  solutions  of  these  problems  are  not,  however,  given. 

It  remains,  in  order  to  close  the  history  of  this  part  of  the  subject,  to  mention 
a  number  of  memoirs  which  although  treating  entirely  new  problems  are  of  far 
less  importance  than  most  of  those  considered  up  to  this  point,  partly  because 
the  solution  is  not  brought  to  a  point  where  it  can  be  of  much  immediate  use, 
and  partly  because  most  of  the  methods  employed  are  such  as  could  not  fail 
to  present  themselves  to  any  one  attacking  these  problems. 

Of  these  the  first  is  a  paper  by  Mathieu'  on  the  vibration  of  an  elliptic 
membrane  (1868),  in  which  the  functions  of  the  elliptic  cylinder  occur  for  the 
first  time. 

This  was  followed  in  the  same  year  by  a  paper  on  closely  allied  subjects  by 
H.  Weber,^  in  which  not  merely  the  case  of  the  complete  ellipse  is  briefly 
considered,  but  also  that  in  which  the  boundary  consists  of  two  arcs  of 
confocal  ellipses  and  two  arcs  of  hyperbolas  confocal  with  them.  The  special 
case  in  which  the  ellipses  and  hyperbolas  become  confocal  parabolas  is  also 
considered,  whereby  the  functions  of  the  parabolic  cylinder  are  for  the  first 
time  introduced. 

In  Mathieu's  "  Cours  de  physique  mathematique "  (1873)  the  problem  of 
the  non-stationary  flow  of  heat  in  an  ellipsoid  is  touched  upon,  and  an 
elaborate  though  not  very  satisfactory  treatment  of  the  special  cases  where 
we  have  ellipsoids  of  revolution  is  given.  New  functions  appear  in  all  of 
these  problems. 

Of  late  years  C.  Baer  has  supplied  a  number  of  missing  links  in  the  chain 
of  problems  here  considered  by  treating  in  succession  the  potential  problem 
for  the  paraboloid  of  revolution,^  the  parabolic  cylinder''  and  the  general 
paraboloid.*  In  the  first  of  these  problems  Bessel's  functions  occur,  as  had 
already  been  stated  by  Mehler,  while  in  the  last  we  find  the  functions  of  the 
elliptic  cylinder.  For  each  of  the  three  systems  of  coordinates  employed  the 
same  author  also  touches  upon  the  more  general  problem  of  the  non-stationary 
flow  of  heat,  in  which  new  functions  occur. 

Except  in  the  case  of  the  anchor  ring  we  have  found  so  far  only  such  solids 
treated   by  our  method  as   are  bounded   by  surfaces  of   the  first  or  second 

1  Liouville's  Journal,  Vol.  XIII. 

2  "  Ueber   die   Inteo;ration   der   partiellen   Differentialgleichung  — — -f- — —  -f- A:%  =  0  ." 

ox-       oy- 

Math.  Ann.,  Vol.  I.     No  physical  problem  is  mentioned  in  this  paper. 

3"Ueber  das  Gleichgewicht  und  die  Bewegung  der  Warme  in  einem  Rotationspara- 
boloid."     Dissertation,  Halle,  1881. 

*  "Die  Funktion  des  parabolischen  Cylinders,"  Gymnasialprogramm  Ciistrin,  1883. 

5  "  Parabolische  Coordinaten,"  Frankfurt,  1888.  See  also  a  paper  by  Greenhill  in  the 
Proc.  Lond.  Math.  See,  Vol.  XIX.,  1889  (read  Dec.  8,  1887).  Also  a  posthumous  paper  by 
Lam6  in  Liouville's  Journal  for  1874,  Vol.  XIX. 


CYCLIDIC    COORDINATES.  273 

degree.  Wangerin^  (1875-76)  considered  in  connection  with  the  theory  of 
the  potential,  more  general  systems  of  curvilinear  coordinates  than  had 
previously  been  used  in  physical  questions,  namely,  cyclidic  coordinates.-  He 
showed,  however,  merely  how  to  break  up  Laplace's  equation  into  three 
ordinary  differential  equations.^ 

An  important  branch  of  our  theory  which  we  have  not  yet  touched  upon 
dates  back  to  the  year  1836,  when  Sturm  published  a  series  of  fundamentally 
important  papers  in  the  first  two  volumes  of  Liouville's  Journal.  The 
physical  question  which  lies  at  the  basis  of  these  papers  is  the  problem  of  the 
flow  of  heat  in  a  heterogeneous  bar.*  The  method  here  employed  depends 
upon  the  fact  that  the  functions  which  occur  are  characterized  by  the  number 
of  times  they  vanish  in  a  certain  interval.  This  same  idea  reappears  in 
Thomson  and  Tait's  Appendix  B  already  referred  to,  but  first  finds  its  full 
expression  in  this  more  general  field  of  the  three  dimensional  potential  in  an 
article  by  Klein  :  "  Ueber  Korper  welche  von  confocalen  Flachen  zweiten 
Grades  begrenzt  sind  "  ^  (1881).  Still  more  recently  (1889-90)  Klein  has  in 
his  lectures  extended  this  theory  to  the  treatment  of  solids  bounded  by  six 
confocal  eyelids,  and  has  indicated  how  all  the  potential  problems  heretofore 
treated  by  our  method  are  special  cases  of  this  one.** 

Of  late  years,  especially  since  the  year  1880,  the  younger  English  mathe- 
maticians have  done  a  vast  amount  of  work  in  the  theory  we  are  here 
considering.  Although  much  of  this  work  is  of  great  value,  hardly  any  of  it 
can  be  regarded  as  being  a  real  development  of  the  method  ;  it  is  rather  an 
application  of  it  to  a  great  variety  of  problems.  We  must  therefore  content 
ourselves  with  giving  a  mere  list  of  a  few  of  the  more  important  of  these 
papers. 

Niven:  On  the  Conduction  of  Heat  in  Ellipsoids  of  Revolution.  Phil. 
Trans.,  1880. 

Niven :  On  the  Induction  of  Electric  Currents  in  Infinite  Plates  and 
Spherical  Shells.     Phil.  Trans.,  1881. 

1  Preisschriften  der  Jablanowski'schen  Gesellschaft,  No.  XVIII.,  and  Crelle's  Journal, 
Vol.  82.  See  also,  concerning  a  still  further  extension,  the  Berliner  Monatsberichten 
for  1878. 

2  Cyclids  are  a  kind  of  surface  of  the  fourth  order  (see  Salmon's  Geom.  of  three  Dimen- 
sions, p.  527).     In  his  first  memoir  Wangerin  considers  only  cyclids  of  revolution. 

3  See  also  a  paper  by  this  author  in  Griinert's  Archiv  for  1873,  where  the  problem  of  the 
equilibrium  of  elastic  solids  of  revolution  is  treated. 

*  The  similar  problem  of  the  vibration  of  a  heterogeneous  string  under  the  action  of  an 
external  force  was  treated  by  Maggi  (Giornale  di  Matematiche,  1880).  Several  special  cases 
are  also  considered  here  in  detail. 

5  Math.  Ann.,  18. 

^  For  an  exposition  of  this  theory  see  the  treatise  :  Ueber  die  Reihenentwickelungen  der 
Potentialtheorie,  Leipsic,  Teubner,  1894,  by  the  writer  of  the  present  chapter. 


274  HISTORICAL    SUMMARY. 

Hicks:  On  Toroidal  Functions.     Phil.  Trans.,  1881. 

Hicks :  On  the  Steady  jMotion  and  Small  A^ibrations  of  a  Hollow  Vortex. 
Phil.  Trans.,  1884,  1885. 

Lamb:  On  Ellipsoidal  Current  Sheets.     Phil.  Trans.,  1887. 

Chree:  The  Equations  of  an  Isotropic  Elastic  Solid  in  Polar  and  Cylin- 
drical Coordinates,  their  Solution  and  Application.  Camb.  Phil.  Soc.  Trans., 
XIV.,  1889. 

Hohson:  On  a  Class  of  Spherical  Harmonics  of  Complex  Degree  with 
Applications  to  Physical  Problems.     Camb.  Phil.  Soc.  Trans.,  XIV.,  1889. 

Chree:  On  some  Compound  Vibrating  Systems.  Camb.  Phil.  Soc.  Trans., 
XV.,  1891. 

Niven:  On  Ellipsoidal  Harmonics.     Phil.  Trans.,  1892. 

The  historical  sketch  we  have  just  given  would  naturally  require  as  a 
supplement  some  account  of  the  work  that  has  been  done  on  the  question  of 
the  convergence  of  the  various  series  which  occur.  This,  however,  would 
carry  us  too  far,  and  we  will  content  ourselves  with  mentioning  the  two 
fundamental  memoirs  by  Dirichlet  in  Crelle's  Journal,  one  in  1829  on 
Fourier's  series,  and  one,  which  has  been  criticised  to  some  extent  by  subse- 
quent mathematicians,  in  1837  on  Laplace's  spherical  harmonic  development. 

Another  subject  which  naturally  presents  itself  here  is  the  theory  of  the 
various  new  functions  we  have  met.  Those  properties  of  these  functions, 
however,  which  the  physicist  needs  have  usually  been  investigated  by  the 
physicists  themselves  in  the  papers  mentioned  above  ;  while  any  thorough 
account  of  the  development  of  the  theory  of  these  functions  would  lead  us 
into  the  vast  region  of  the  modern  theory  of  linear  differential  equations. 

We  will  therefore  close  by  merely  giving  a  list  of  books  which  will  be 
found  useful  by  those  wishing  to  continue  their  study  of  the  subject  further. 

We  begin  with  the  books  relating  directly  to  physical  questions  : 

Fourier:  Theorie  Analytique  de  la  Chaleur,  1822. 

Lame  :  Leqons  sur  les  Eonctions  inverses  des  Transcendantes  et  les  Surfaces 
isothermes,  1857. 

Lame:  Leqons  sur  les  Coordonnees  Curvilignes  et  leurs  diverses  Applica- 
tions, 1859. 

Mathieu :  Cours  de  Physique  Mathematique,  1873. 

Riemann:  Partielle  Differentialgleichungen,  und  deren  Anwendung  auf 
physikalische  Fragen  (edited  by  Hattendorf),  third  edition,  1882. 

F.  Neumann:  Theorie  des  Potentials  und  der  Kugelfunktionen  (edited  by 
C.  Neumann),  1887. 

Thomson  and  Tait :  Natural  Philosophy,  second  edition,  1879. 

Bayleigh:  Theory  of  Sound,  1877. 

Basset:  Hydrodynamics,  1888. 

Love :  Theory  of  Elasticity,  1892. 


BOOKS    OF    REFERENCE.  275 

Heine :  Handbuch  der  Kugelfuiiktionen  (second  edition),  1878-81. 

Ferrers :  Spherical  Harmonics,  1881. 

Haentzschel :  Reduction  der  Potentialgleichung  auf  gewohnliclie  Differential- 
gleichungen,  1893. 

These  last  three  books  would  also  belong  in  the  following  list  of  books 
relating  to  the  theory  of  the  various  functions  we  use  : 

Todhunter :  The  Functions  of  Laplace,  Lame  and  Bessel,  1875. 

Lommel:  Studien  liber  die  Bessel'schen  Punktionen,  1868. 

F.  Neumann :  Beitrage  zur  Theorie  der  Kugelfunktionen,  1878. 

And  finally  concerning  the  question  of  convergence  : 

C.  Neumann:  tjber  die  nach  Kreis-,  Kugel-  und  Cylinder-Functionen 
fortschreitenden  Entwickelungen,  1881. 


APPENDIX 


TABLES. 


Table  I.,  a  table  of  Surface  Zonal  Harmonics  (Legendrians),  gives  the  values 
of  the  first  seven  Harmonics  P^  (cos  9),  P^  (cos  $),  •••  P^  (cos  6)  for  the  argument 
6  in  degrees.  It  is  taken  from  the  Philosophical  Magazine  for  December, 
1891,  and  was  computed  by  Messrs.  C.  E.  Holland,  P.  E.  James,  and  C.  G. 
Lamb,  under  the  direction  of  Professor  John  Perry. 

Table  II.,  a  table  of  Surface  Zonal  Harmonics  (Legendrians),  gives  the 
values  of  the  first  seven  Harmonics  Pi (x),  P^{x),  ••  •  P^ (x)  for  the  argument  x. 
It  is  reduced  from  the  Tables  of  Legendrian  Functions  computed  under  the 
direction  of  Dr.  J.  W.  L.  Glaisher,  and  published  in  the  Report  of  the  British 
Association  for  the  Advancement  of  Science  for  the  year  1879. 

Table  III.,  the  table  of  Hyperbolic  Functions,  gives  the  values  of  e-'^,  e'~^, 
sinlix,  cosh  if,  and  gdx  (Gudermannian  of  x)  for  values  of  x  from  0.00  to  1.00; 
and  the  values  of  logsinhx  and  log  cosh  x  for  values  of  x  from  1.00  to  10.0. 
The  values  of  gd  a-,  log  sinh  x,  and  log  cosh  x  are  taken  from  the  Mathematical 
Tables  prepared  by  Professor  J.  M.  Peirce  (Boston:  Ginn  &  Co.). 

The  log  sinh  a;  and  log  cosh  a-  for  values  of  x  between  0.00  and  1.00  can  be 
obtained  from  the  values  given  for  the  Gudermannian  of  x  in  the  table  by  the 
aid  of  the  relations 

log  sinh  X.  =  log  tan  (gd  x) 

log  cosh  X  =  log  sec  (gd  x). 

Table  IV.  gives  the  first  twelve  roots  of  Jq  (x)  =  0  and  Ji  (x)  =  0  each 
divided  by  tt.  The  table  is  taken  from  Lord  Rayleigh's  Sound,  Vol.  I., 
page  274,  and  is  due  to  Professor  Stokes,  Camb.  Phil.  Trans.,  Vol.  IX., 
page  186. 

Table  V.  gives  the  first  nine  roots  of  Jq (x)  =  0,  Ji (x)  =  0,  -■  •  J^ (x)  =  0. 
The  table  is  taken  from  Rayleigh's  Sound,  Vol.  I.,  page  274,  and  is  due  to 
Professor  J.  Bourget,  Ann.  de  I'Ecole  Normale,  T.  III.,  1866,  page  82. 

Table  VI.,  the  table  of  Bessel's  Functions,  gives  the  values  of  the  Bessel's 
Functions  Jo  (^)  and  J^  (x)  for  the  argument  x  from  ic  =  0  to  ic  =  15.  It  is 
taken  from  Rayleigh's  Sound,  Vol.  I.,  page  265,  and  from  Lommel's  Bessel'sche 
Functionen. 


278 


APPENDIX. 

TABLE   I.  —  Surface  Zonal  Harmonics. 


e 

Pi  (cos  e) 

Pa  (cos  d) 

PzicosO) 

Pi{cose) 

P5(cos^) 

P6(cose) 

P7(COS6) 

0° 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1 

.9998 

.9995 

.9991 

.9985 

.9977 

.9967 

.9955 

2 

.9994 

.9982 

.9963 

.9939 

.9909 

.9872 

.9829 

3 

.9986 

.9959 

.9918 

.9863 

.9795 

.9713 

.9617 

4 

.9976 

.9927 

.9854 

.9758 

.9638 

.9495 

.9329 

5 

.9962 

.9886 

.9773 

.9623 

.9437 

.9216 

.8961 

6 

.9945 

.9836 

.9674 

.9459 

.9194 

.8881 

.8522 

7 

.9925 

.9777 

.9557 

.9267 

.8911 

.8476 

.7986 

S 

.9903 

.9709 

.9423 

.9048 

.8589 

.8053 

.7448 

9 

.9877 

.9633 

.9273 

.8803 

.8232 

.7571 

.6831 

10 

.9848 

.9548 

.9106 

.8532 

.7840 

.7045 

.6164 

11 

.9816 

.9454 

.8923 

.8238 

.7417 

.6483 

.5461 

12 

.9781 

.9352 

.8724 

.7920 

.6966 

.5892 

.4732 

13 

.9744 

.9241 

.8511 

.7582 

.6489 

.5273 

.3940 

14 

.9703 

.9122 

.8283 

.7224 

.5990 

.4635 

.3219 

15 

.9659 

.8995 

.8042 

.6847 

.5471 

.3982 

.2454 

16 

.9613 

.8860 

.7787 

.6454 

.4937 

.3322 

.1699 

17 

.9563 

.8718 

.7519 

.6046 

.4391 

.2660 

.0961 

18 

.9511 

.8568 

.7240 

.5624 

.3836 

.2002 

.0289 

19 

.9455 

.8410 

.6950 

.5192 

.3276 

.1347 

—.0443 

20 

.9397 

.8245 

.6649 

.4750 

.2715 

.0719 

-.1072 

21 

.9336 

.8074 

.6338 

.4300 

.2156 

.0107 

-.1662 

22 

.9272 

.7895 

.6019 

.3845 

.1602 

—.0481 

—.2201 

23 

.9205 

.7710 

.5692 

.3386 

.1057 

—.1038 

—.2681 

24 

.9135 

.7518 

.5357 

.2926 

.0525 

—.1559 

—.3095 

25 

.9063 

.7321 

.5016 

.2465 

.0009 

—.2053 

—.3463 

26 

.8988 

.7117 

.4670 

.2007 

-.0489 

—.2478 

—.3717 

27 

.8910 

.6908 

.4319 

.1553 

-.0964 

—.2869 

—.3921 

28 

.8829 

.6694 

.3964 

.1105 

-.1415 

-.3211 

—.4052 

29 

.8746 

.6474 

.3607 

.0665 

-.1839 

-.3503 

—.4114 

30 

.8660 

.6250 

.3248 

.0234 

—.2233 

-.3740 

-.4101 

31 

.8572 

.6021 

.2887 

—.0185 

—.2595 

—.3924 

—.4022 

32 

.8480 

.5788 

.2527 

—.0591 

—.2923 

—.4052 

-.3876 

33 

.8387 

.5551 

.2167 

—.0982 

—.3216 

—.4126 

-.3670 

34 

.8290 

.5310 

.1809 

-.1357 

—.3473 

-.4148 

-.3409 

35 

.8192 

.5065 

.1454 

—.1714 

—.3691 

—.4115 

—.3096 

36 

.8090 

.4818 

.1102 

-.2052 

—.3871 

—.4031 

-.2738 

37 

.7986 

.4567 

.0755 

-.2370 

—.4011 

—.3898 

—.2343 

38 

.7880 

.4314 

.0413 

-.2666 

—.4112 

—.3719 

—.1918 

39 

.7771 

.4059 

.0077 

—.2940 

—.4174 

—.3497 

—.1469 

40 

.7660 

.3802 

—.0252 

-.3190 

—.4197 

-.3234 

—.1003 

41 

.7547 

.3544 

-.0574 

—.3416 

-.4181 

-.2938 

—.0534 

42 

.7431 

.3284 

-.0887 

—.3616 

-.4128 

-.2611 

-.0065 

43 

.7314 

.3023 

—.1191 

—.3791 

—.4038 

-.2255 

.0398 

44 

.7193 

.2762 

—.1485 

—.3940 

—.3914 

-.1878 

.0846 

45° 

.7071 

.2500 

—.1768 

—.4062 

-.3757 

-.1485 

.1270 

TABLE    I. 


APPENDIX. 
Surface  Zonal  Harmonics. 


279 


6 

Pi  (cos  e) 

P2(cose) 

Pa  (cos  6*) 

Pi  (cos  6) 

Ps  (cos  6) 

Pe  (cos  d) 

P^  (cos  6) 

45° 

.7071 

.2500 

-.1768 

—.4062 

—.3757 

—.1485 

.1270 

46 

.6947 

.2238 

—.2040 

-.41.58 

—.3568 

—.1079 

.1666 

47 

.6820 

.1977 

—.2300 

—.4252 

—.33.50 

—.0645 

.2054 

48 

.6691 

.1716 

—.2547 

—.4270 

—.3105 

—.0251 

.2349 

49 

.6561 

.  1456 

—.2781 

—.4286 

—.2836 

.0161 

.2627 

50 

.6428 

.1198 

—.3002 

—.4275 

—.2545 

.0563 

.2854 

51 

.6293 

.0941 

—.3209 

—.4239 

—.2235 

.0954 

.3031 

52 

.6157 

.0686 

—.3401 

-.4178 

—.1910 

.1326 

.3153 

53 

.6018 

.0433 

—  3578 

—.4093 

—.1571 

.1677 

.3221 

54 

.5878 

.0182 

—.3740 

-.3984 

-.1223 

.2002 

.3234 

55 

.5736 

—.0065 

-.3886 

-.3852 

-.0868 

.2297 

.3191 

56 

.5592 

—.0310 

—.4016 

-.3698 

-.0510 

.2559 

.3095 

57 

.5446 

—.0551 

-.4131 

-.3524 

-.0150 

.2787 

.2949 

58 

.5299 

-.0788 

—.4229 

-.3331 

.0206 

.2976 

.2752 

59 

.5150 

—.1021 

-.4310 

-.3119 

.0557 

.3125 

.2511 

60 

.5000 

—.1250 

—.4375 

—.2891 

.0898 

.3232 

.2231 

61 

.4848 

—.1474 

—.4423 

—.2647 

.1229 

.3298 

.1916 

62 

.4695 

—.1694 

—.4455 

—.2390 

.1545 

.3321 

.1571 

63 

.4540 

—.1908 

—.4471 

—.2121 

.1844 

.3302 

.1203 

64 

.4384 

-.2117 

—.4470 

— .1S41 

.2123 

.3240 

.0818 

65 

.4226 

—.2321 

—.4452 

—.1552 

.2381 

.3138 

.0422 

66 

.4067 

—.2518 

-.4419 

—.1256 

.2615 

.2996 

.0021 

67 

.3907 

—.2710 

—.4370 

—.0955 

.2824 

.2S19 

-.0375 

68 

.3746 

—.2896 

—.4305 

—.0650 

.3005 

.2605 

-.0763 

69 

.3584 

—.3074 

—.4225 

—.0344 

.3158 

.2361 

-.1135 

70 

.3420 

—.3245 

—.4130 

—.0038 

.3281 

.2089 

-.1485 

71 

.3256 

—.3410 

—.4021 

.0267 

.3373 

.1786 

-.1811 

72 

.3090 

—.3568 

-.3898 

.0568 

.3434 

.1472 

—.2099 

73 

.2924 

—.3718 

—.3761 

.0864 

.3463 

.1144 

—.2347 

74 

.2756 

—.3860 

—.3611 

.1153 

.3461 

.0795 

-.2559 

75 

.2588 

—.3995 

—.3449 

.1434 

.3427 

.0431 

—.2730 

76 

.2419 

—.4112 

-.3275 

.1705 

.3362 

.0076 

-.2848 

77 

.2250 

—.4241 

—.3090 

.1964 

.3267 

-.0284 

—.2919 

78 

.2079 

—.4352 

—.2894 

.2211 

.3143 

—.0644 

—.2943 

79 

.1908 

—.4454 

-.2688 

.2443 

.2990 

—.0989 

—.2913 

SO 

.1736 

—.4548 

—.2474 

.2659 

.2810 

—.1321 

-.2835 

81 

.1564 

-.4633 

-.2251 

.2859 

.2606 

— .  1635 

—.2709 

82 

.1392 

—.4709 

—.2020 

.3040 

.2378 

—.1926 

—  2536 

S3 

.1219 

—.4777 

-.1783 

.3203 

.2129 

—.2193 

—.2321 

S4 

.1045 

-.4836 

—.1539 

.3345 

.1861 

-.2431 

—.2067 

85 

.0872 

-.4886 

—.1291 

.3468 

.1577 

-.2638 

—.1779 

86 

.0698 

—.4927 

-.1038 

.3569 

.1278 

-.2811 

— .H60 

87 

.0523 

—.4959 

—.0781 

.3648 

.0969 

-.2947 

—.1117 

88 

.0349 

-.4982 

—.0522 

.3704 

.0651 

—.3045 

—.0735 

89 

.0175 

—.4995 

—.0262 

.3739 

.0327 

—.3105 

—.0381 

90° 

.0000 

-.5000 

.0000 

.3750 

.0000 

-.3125 

.0000 

280 


TABLE   II. 


APPENDIX. 
Surface  Zoxal  Harmoxics. 


X 

Pi(.r) 

Po(a.-) 

-PsCx) 

P4(x) 

A(x) 

PgW 

P,(x) 

0.00 

0.0000 

—.5000 

0.0000 

0.3750 

0.0000 

—.3125 

0.0000 

.01 

.0100 

—.4998 

—.0150 

.3746 

.0187 

—.3118 

—.0219 

.02 

.0200 

—.4994 

—.0300 

.3735 

.0374 

—.3099 

—.0436 

.03 

.0300 

—.4986 

—.0449 

.3716 

.0560 

—.3066 

—.0651 

.04 

.0400 

—.4976 

—.0598 

.3690 

.0744 

—.3021 

—.0862 

.05 

.0500 

—.4962 

—.0747 

.3657 

.0927 

—.2962 

—.1069 

.06 

.0600 

—.4946 

—.0895 

.3616 

.1106 

—.2891 

—.1270 

.07 

.0700 

—.4926 

—.1041 

.3567 

.1283 

—.2808 

—.1464 

.OS 

.0800 

—.4904 

—.1187 

.3512 

.1455 

—.2713 

—.1651 

.09 

.0900 

—.4878 

-.1332 

.3449 

.1624 

—.2606 

-.1828 

.10 

.1000 

—.4850 

-.1475 

.3379 

.1788 

—.2488 

-.1995 

.11 

.1100 

—.4818 

—.1617 

.3303 

.1947 

—.2360 

—.2151 

.12 

.1200 

-.4784 

—.1757 

.3219 

.2101 

—.2220 

—.2295 

.13 

.1300 

—.4746 

-.1895 

.3129 

.2248 

—.2071 

—.2427 

.14 

.1400 

-.4706 

—.2031 

.3032 

.2389 

—.1913 

—.2545 

.15 

.1500 

—.4662 

—.2166 

.2928 

.2523 

—.1746 

—.2649 

.16 

.1600 

—.4616 

—.2298 

.2819 

.2650 

-.1572 

—.2738 

.17 

.1700 

—.4566 

—.2427 

.2703 

.2769 

-.1389 

—.2812  ; 

.18 

.1800 

-.4514 

—.2554 

.2581 

.2880 

—.1201 

-.2870 

.19 

.1900 

-.4458 

—.2679 

.2453 

.2982 

—.1006 

—.2911  1 

.20 

.2000 

—.4400 

—.2800 

.2320 

.3075 

-.0806 

-.2935 

.21 

.2100 

-.4338 

—.2918 

.2181 

.3159 

-.0601 

—.2943 

.22 

.2200 

—.4274 

—.3034 

.2037 

.3234 

—.0394 

-.2933 

.23 

.2300 

—.4206 

—.3146 

.1889 

.3299 

-.0183 

—.2906 

.24 

.2400 

—.4136 

—.3254 

.1735 

.3353 

.0029 

—.2861 

.25 

.2500 

—.4062 

—.3359 

.1577 

.3397 

.0243 

-.2799 

.26 

.2600 

-.3986 

—.3461 

.1415 

.3431 

.0456 

—.2720 

.27 

.2700 

—.3906 

—.3558 

.1249 

.3453 

.0669 

-.2625 

.28 

.2800 

-.3824 

-.3651 

.1079 

.3465 

.0879 

-.2512 

.29 

.2900 

-.3738 

—.3740 

.0906 

.3465 

.1087 

-.2384 

.30 

.3000 

-.3650 

—.3825 

.0729 

.3454 

.1292 

—.2241 

.31 

.3100 

-.3558 

—.3905 

.0550 

.3431 

.1492 

-.2082 

.32 

.3200 

—.3464 

—.3981 

.0369 

.3397 

.1686 

—.1910 

.33 

.3300 

-.3366 

—.4052 

.0185 

.3351 

.1873 

-.1724 

.34  ■ 

.3400 

—.3266 

—.4117 

—.0000 

.3294 

.2053 

—.1527 

.35 

.3500 

—.3162 

—.4178 

—.0187 

.3225 

.2225 

—.1318 

.36 

.3600 

—.3056 

—.4234 

—.0375 

.3144 

.2388 

-.1098 

.37 

.3700 

—.2946 

—.4284 

—.0564 

.3051 

.2540 

—.0870 

.38 

.3800 

—.2834 

—.4328 

—.0753 

.2948 

.2681 

—.0635 

.39 

.3900 

-.2718 

—.4367 

—.0942 

.2833 

.2810 

—.0393 

.40 

.4000 

—.2600 

—.4400 

—.1130 

.2706 

.2926 

—.0146 

.41 

.4100 

-.2478 

—.4427 

—.1317 

.2569 

.3029 

.0104 

.42 

.4200 

—.2354 

—.4448 

—.150+ 

.2421 

.3118 

.0356 

.43 

.4300 

—.2226 

—.4462 

—.1688 

.2263 

.3191 

.0608 

.44 

.4400 

—.2096 

—.4470 

—.1870 

.2095 

.3249 

.0859 

.45 

.4500 

—.1962 

—.4472 

—.2050 

.1917 

.3290 

.1106 

.46 

.4600 

—.1826 

—.4467  - 

—.2226 

.1730 

.3314 

.1348 

.47 

.4700 

—.1686 

—.4454 

—.2399 

.1534 

.3321 

.1584 

.48 

.4800 

—.1544 

—.4435 

—.2568 

.1330 

.3310 

.1811 

.49 

.4900 

—.1398 

—.4409 

—.2732 

.1118 

.3280 

.2027 

.50 

.5000 

—.1250 

-.4375 

-.2891 

.0898 

.3232 

.2231 

appe:*dix. 
TABLE    II.  —  Surface  Zonal  Harmonics. 


281 


X 

Pi  (x) 

P-2  (X) 

Pz{x) 

Pi  ic) 

Pb  (x) 

Po(x) 

P7(X) 

.50 

.5000 

—.1250 

—.4375 

—.2891 

.0898 

.3232 

.2231 

.51 

.5100 

—.1098 

—.4334 

—.3044 

.0673 

.3166 

.2422 

52 

.5200 

—.0944 

—.4285 

—.3191 

.0441 

.3080 

.2596 

.5.'5 

.5300 

—.0786 

—.4228 

—.3332 

.0204 

.2975 

.2753 

.54 

.5400 

-.0626 

—.4163 

—.3465 

—.0037 

.2851 

.2891 

.55 

.5500 

—.0462 

-.4091 

—.3590 

—.0282 

.2708 

.3007 

.56 

.5600 

—.0296 

—.4010 

-.3707 

—.0529 

.2546 

.3102 

•  57 

.5700 

—.0126 

—.3920 

-.3815 

-.0779 

.2366 

.3172 

.58 

.5800 

.0046 

-.3822 

-.3914 

—.1028 

.2168 

.3217 

.59 

.5900 

.0222 

—.3716 

—.4002 

-.1278 

.1953 

.3235 

.60 

.6000 

.0400 

-.3600 

—.4080 

—.1526 

.1721 

.3226 

.61 

.6100 

.0582 

-.3475 

—.4146 

—.1772 

.1473 

.3188 

.62 

.6200 

.0766 

-.3342 

—.4200 

—.2014 

.1211 

.3121 

.63 

.6300 

.0954 

—.3199 

—.4242 

—.2251 

.0935 

.3023 

.64 

.6400 

.1144 

—.3046 

—.4270 

—.2482 

.0646 

.2895 

.65 

.6500 

.1338 

—.2884 

—.4284 

—.2705 

.0347 

.2737 

.66 

.6600 

.1.534 

—.2713 

-.4284 

—.2919 

.0038 

.2548 

.67 

.6700 

.1734 

—.2531 

—.4268 

—.3122 

-.0278 

.2329 

.68 

.6800 

.1936 

—.2339 

—.4236 

—.3313 

-.0601 

.2081 

.69 

.6900 

.2142 

—.2137 

—.4187 

—.3490 

—.0926 

.1805 

.70 

.7000 

.2350 

—.1925 

-.4121 

—.3652 

—.1253 

.1502 

.71 

.7100 

.2562 

—.1702 

—.4036 

—.3796 

—.1578 

.1173 

.72 

.7200 

.2776 

—.1469 

—.3933 

—.3922 

—.1899 

.0822 

.73 

.7300 

.2994 

—.1225 

—.3810 

—.4026 

—.2214 

.0450 

.74 

.7400 

.3214 

—.0969 

—.3666 

-.4107 

—.2518 

.0061 

75 

.7500 

.3438 

—.0703 

—.3501 

—.4164 

-.2808 

—.0342 

'76 

.7600 

.3664 

—.0426 

•  -.3314 

—.4193 

—.3081 

—.0754 

.77 

.7700 

.3894 

—.0137 

—.3104 

—.4193 

—.3333 

-.1171 

.78 

.7800 

.4126 

.0164 

—.2871 

—.4162 

—.3559 

-.1588 

.79 

.7900 

.4362 

.0476 

-.2613 

—.4097 

—.3756 

—.1999 

.80 

.8000 

.4600 

.0800 

—.2330 

—.3995 

—.3918 

—.2.397 

.81 

.8100 

.4842 

.1136 

—.2021 

—.3855 

—.4041 

-.2774 

.82 

.8200 

.5086 

.1484 

-.1685 

—.3674 

-.4119 

—.3124 

.83 

.8300 

.5334 

.1845 

—.1321 

—.3449 

-.4147 

-.3437 

.84 

.8400 

.5584 

.2218 

—.0928 

—.3177 

—.4120 

-.3703 

.85 

.8500 

.5838 

.2603 

-.0506 

—.2857 

—.4030 

—.3913 

.86 

.8600 

.6094 

.3001 

—.0053 

-.2484 

—.3872 

—.4055 

.87 

.8700 

.6354 

.3413 

.0431 

—.2056 

—.3638 

—.4116 

.88 

.8800 

.6616 

.3837 

.0947 

—.1570 

—.3322 

—.4083 

.89 

.8900 

.6882 

.4274 

.1496 

-.1023 

—.2916 

—.3942 

.90 

.9000 

.7150 

.4725 

.2079 

—.0411 

—.2412 

-.3678 

.91 

.9100 

.7422 

.5189 

.2698 

.0268 

—.1802 

— .3274 

.9200 

.7696 

.5667 

.3352 

.1017 

— .1077- 

-.2713 

.93 

.9300 

.7974 

.6159 

.4044 

.1842 

—.0229 

-.1975 

.94 

.9400 

.8254 

.6665 

.4773 

.2744 

.0751 

-.1040 

.95 

.9500 

.8538 

.7184 

.5541 

.3727 

.1875 

.0112 

■96 

.9600 

.8824 

.7718 

.6349 

.4796 

.3151 

.1506 

.97 

.9700 

.9114 

.8267 

.7198 

.5954 

.4590 

.3165 

.98 

.9800 

.9406 

.8830 

.8089 

.7204 

.6204 

.5115 

.99 

.9900 

.9702 

.9407 

.9022 

.8552 

.8003 

.7384 

1.00 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

282 


APPENDIX. 

TABLE    III.  —  Hyperbolic  Functioxs. 


X 

e.r 

e-x 

sinhx 

cosh  X 

gdx 

0.00 

1.0000 

1.0000 

0.0000 

1.0000 

0?0000 

.01 

1.0100 

0.9900 

.0100 

1.0000 

0.5729 

.02 

1.0202 

.9802 

.0200 

1.0002 

1.1458 

.03 

1.0305 

.9704 

.0300 

1.0004 

1.7186 

.04 

1.0408 

.9608 

.0400 

1.0008 

2.2912 

.05 

1.0513 

.9512 

.0500 

1.0012 

2.8636 

.06 

1.0618 

.9418 

.0600 

1.0018 

3.4357 

.07 

1.0725 

.9324 

.0701 

1.0025 

4.0074 

.OS 

1.0S33 

.9231 

.0801 

1.0032 

4.5788 

.09 

1.0942 

.9139 

.0901 

1.0040 

5.1497 

.10 

1.1052 

.9048 

.1002 

1.0050 

5.720 

.11 

1.1163 

.8958 

.1102 

1.0061 

6.290 

.12 

1.1275 

.8869 

.1203 

1.0072 

6.859 

.13 

1.13SS 

.8781 

.1304 

1.0085 

7.428 

.14 

1.1503 

.8694 

.1405 

1.0098 

7.995 

.15 

1.1618 

.8607 

.1506 

1.0113 

8.562 

.16 

1.1735 

.8521 

.1607 

1.0128 

9.128 

.17 

1.1853 

.8437 

.1708 

1.0145 

9.694 

.IS 

1.1972 

.8353 

.1810 

1.0162 

10.258 

.19 

1.2092 

.8270 

.1911 

1.0181 

10.821 

.20 

1.2214 

.8187 

.2013 

1.0201 

11.384 

.21 

1.2337 

.8106 

.2115 

1.0221 

11.945 

.22 

1.2461 

.8025 

.2218 

1.0243 

12.505 

!23 

1.2586 

.7945 

.2320 

1.0266 

13.063 

.24 

1.2712 

.7866 

.2423 

1.0289 

13.621 

.25 

1.2840 

.7788 

.2526 

1.0314 

14.177 

.26 

1.2969 

.7711 

.2629 

1.0340 

14.732 

.27 

1.3100 

.7634 

.2733 

1.0367 

15.285 

.28 

1.3231 

.7558 

.2837 

1.0395 

15.837 

.29 

1.3364 

.7483 

.2941 

1.0423 

16.388 

.30 

1.3499 

.7408 

.3045 

1.0453 

16.937 

.31 

1.3634 

.7334 

.3150 

1.0484 

17.484 

.32 

1.3771 

.7261 

.3255 

1.0516 

18.030 

32, 

1.3910 

.7189 

.3360 

1.0549 

18.573 

.34 

1.4049 

.7118 

.3466 

1.0584 

19.116 

.35 

1.4191 

.7047 

.3572 

1.0619 

19.656 

.36 

1.4333 

.6977 

.3678 

1.0655 

20.195 

.37 

1.4477 

.6907 

.3785 

1.0692 

20.732 

.38 

1.4623 

.6839 

.3892 

1.0731 

21.267 

.39 

1.4770 

.6771 

.4000 

1.0770 

21.800 

.40 

1.4918 

.6703 

.4108 

1.0811 

22.331 

.41 

1.5068 

.6636 

.4216 

1.0852 

22.859 

.42 

1.5220 

.6570 

.4325 

1.0895 

23.386 

.43 

1.5373 

.6505 

.4434 

1.0939 

23.911 

.44 

1.5527 

.6440 

.4543 

1.0984 

24.434 

.45 

1.5  683 

.6376 

.4653 

1.1030 

24.955 

.46 

1.5841 

.6313 

.4764 

1.1077 

25.473 

.47 

1.6000 

.6250 

.4875 

1.1125 

25.989 

.48 

1.6161 

.6188 

.4986 

1.1174 

26.503 

.49 

1.6323 

.6126 

.5098 

1.1225 

27.015 

0.50 

1.6487 

0.6065 

0.5211 

1.1276 

27?524 

APPENDIX. 

TABLE    III.  —  Hyperbolic  Functiox> 


283 


X 

e-r 

e— 

sinh  X 

cosh  X 

gdx 

0.50 

1.6487 

0.6065 

0.5211 

1.1276 

27?524 

.51 

1.6653 

.6005 

.5324 

1.1329 

28.031 

.52 

1.6820 

.5945 

.5438 

1.1383 

28.535 

.53 

1.6989 

.5886 

.5552 

1.1438 

29.037 

.54 

1.7160 

.5827 

.5666 

1.1494 

29.537 

.55 

1.7333 

.5770 

.5782 

1.1551 

30.034 

.56 

1.7507 

.5712 

.5897 

1.1609 

30.529 

.57 

1.76S3 

.5655 

.6014 

1.1669 

31.021 

.58 

1.7860 

.5599 

.6131 

1.1730 

31.511 

.59 

1.8040 

.5543 

.6248 

1.1792 

31.998 

.60 

1.8221 

.5488 

.6367 

1.1855 

32.483 

.61 

1.8404 

.5433 

.6485 

1.1919 

32.965 

.62 

1.8589 

.5379 

.6605 

1.1984 

33.444 

.63 

1.8776 

.5326 

.6725 

1.2051 

33.921 

.64 

1.8965 

.5273 

.6846 

1.2119 

34.395 

.65 

1.9155 

.5220 

.6967 

1.2188 

34.867 

.66 

1.9348 

.5169 

.7090 

1.2258 

35.336 

.67 

1.9542 

.5117 

.7213 

1.2330 

35.802 

.68 

1.9739 

.5066 

.7336 

1.2402 

36.265 

.69 

1.9937 

.5016 

.7461 

1.2476 

36.726 

.70 

2.0138 

.4966 

.7586 

1.2552 

37.183 

.71 

2.0340 

.4916 

.7712 

1.2628 

37.638 

.72 

2.0544 

.4867 

.7838 

1.2706 

38.091 

.73 

2.0751 

.4819 

.7966 

1.2785 

38.540 

.74 

2.0959 

.4771 

.8094 

1.2865 

38.987 

.75 

2.1170 

.4724 

.8223 

1.2947 

39.431 

.76 

2.1383 

.4677 

.8353 

1.3030 

39.872 

.77 

2.1598 

.4630 

.8484 

1.3114 

40.310 

.78 

2.1815 

.4584 

.8615 

•  1.3199 

40.746 

.79 

2.2034 

.4538 

.8748 

1.3286 

41.179 

.80 

2.2255 

.4493 

.8881 

1.3374 

41.608 

.81 

2.2479 

.4449 

.9015 

1.3464 

42.035 

.82 

2.2705 

.4404 

.9150 

1.3555 

42.460 

.83 

2.2933 

.4360 

.9286 

1.3647 

42.881 

.84 

2.3164 

.4317 

.9423 

1.3740 

43.299 

.85 

2.3396 

.4274 

.9561 

1.3835 

43.715 

.86 

2.3632 

.4232 

.9700 

1.3932 

44.128 

.87 

2.3869 

.4190 

.9840 

1.4029 

44.537 

.88 

2.4109 

.4148 

.9981 

1.4128 

44.944 

.89 

2.4351 

.4107 

1.0122 

1.4229 

45.348 

.90 

2.4596 

.4066 

1.0265 

1.4331 

45.7.^0 

.91 

2.4843 

.4025 

1.0409 

1.4434 

46.148 

.92 

2.5093 

.3985 

1.0554 

1.4539 

46.544 

.93 

2.5345 

.3946 

1.0700 

1.4645 

46.936 

.94 

2.5600 

.3906 

1.0847 

1.4753 

47.326 

.95 

2.5857 

.3867 

1.0995 

1.4862 

47.713 

.96 

2.6117 

.3829 

1.1144 

1.4973 

48.097 

.97 

2.6379 

.3791 

1.1294 

1.5085 

48.478 

.98 

2.6645 

.3753 

1.1446 

1.5199 

48.857 

.99 

2.6912 

.3716 

1.1598 

1.5314 

49.232 

1.00 

2.7183 

0.3679 

1.1752 

1.5431 

49/505 

284 


APPENDIX. 

TABLE    III.  —  Hyperbolic  Functions 


X 

Lsiuhu; 

I  cosh  X 

X 

I  sinli  X 

I  cosh  X 

X 

I  sinh  X 

I  cosh  X 

1.00 

0.0701 

0.1884 

1.50 

0.3282 

0.3715 

2.00 

0.5595 

0.5754 

1.01 

.0758 

.1917 

].51 

.3330 

.3754 

2.01 

.5640 

.5796 

1.02 

.0815 

.  1950 

1.52 

.3378 

.3794 

2.02 

.5685 

.5838 

1.03 

.0871 

.1984 

1.53 

.3426 

.3833 

2.03 

.5730 

.5880 

1.04 

.0927 

.2018 

1.54 

.3474 

.3873 

2.04 

.5775 

.5922 

1.05 

.0982 

.2051 

1.55 

.3521 

.3913 

2.05 

.5820 

.5964 

1.06 

.1038 

.2086 

1.56 

.3569 

.3952 

2.06 

.5865 

.6006 

1.07 

.1093 

.2120 

1.57 

.3616 

.3992 

2.07 

.5910 

.6048 

l.OS 

.1148 

.2154 

].5S 

.3663 

.4032 

2.08 

.5955 

.6090 

1.09 

.1203 

.2189 

1.59 

.3711 

.4072 

2.09 

.6000 

.6132 

1.10 

.1257 

.2223 

1.60 

.3758 

.4112 

2.10 

.6044 

.6175 

1.11 

.1311 

.2258 

1.61 

.3805 

.4152 

2.11 

.6089 

.6217 

1.12 

.1365 

.2293 

1.62 

.3852 

.4192 

2.12 

.6134 

.6259 

1.13 

.1419 

.2328 

1.63 

.3899 

.4232 

2.13 

.6178 

.6301 

1.14 

.1472 

.2364 

1.64 

.3946 

.4273 

2.14 

.6223 

.6343 

1.15 

.1525 

.2399 

1.65 

.3992 

.4313 

2.15 

.6268 

.6386 

1.16 

.1578 

.2435 

1.66 

.4039 

.4353 

2.16 

.6312 

.6428 

1.17 

.1631 

.2470 

1.67 

.4086 

.4394 

2.17 

.6357 

.6470 

1.18 

.1684 

.2506 

1.68 

.4132 

.4434 

2.18 

.6401 

.6512 

1.19 

.1736 

.2542 

1.69 

.4179 

.4475 

2.19 

.6446 

.6555 

1.20 

.1788 

.2578 

1.70 

.4225 

.4515 

2.20 

.6491 

.6597 

1.21 

.1840 

.2615 

1.71 

.4272 

.4556 

2.21 

.6535 

.6640 

1.22 

.1892 

.2651 

1.72 

.4318 

.4597 

2.22 

.6580 

.6682 

1.23 

.1944 

.2688 

1.73 

.4364 

.4637 

2.23 

.6624 

.6724 

1.24 

.1995 

.2724 

1.74 

.4411 

.4678 

2.24 

■  .6668 

.6767 

1.25 

.2046 

.2761 

1.75 

.4457 

.4719 

2.25 

.6713 

.6809 

1.26 

.2098 

.2798 

1.76 

.4503 

.4760 

2.26 

.6757 

.6852 

1.27 

.2148 

.2835 

1.77 

.4549 

.4801 

2.27 

.6802 

.6894 

1.28 

.2199 

.2872 

1.78 

.4595 

.4842 

2.28 

.6846 

.6937 

1.29 

.2250 

.2909 

1.79 

.4641 

.4883 

2.29 

.6890 

.6979 

1.30 

.2300 

.2947 

l.SO 

.4687 

.4924 

2.30 

.6935 

.7022 

1.31 

.2351 

.2984 

1.81 

.4733 

.4965 

2.31 

.6979 

.7064 

1.32 

.2401 

.3022 

1.82 

.4778 

.5006 

2.32 

.7023 

.7107 

1.33 

.2451 

.3059 

1.83 

.4824 

.5048 

2.33 

.7067 

.7150 

1.34 

.2501 

.3097 

1.84 

.4870 

.5089 

2.34 

.7112 

.7192 

1.35 

.2551 

.3135 

1.S5 

.4915 

.5130 

2.35 

.7156 

.7235 

1.36 

.2600 

.3173 

1.86 

.4961 

.5172 

2.36 

.7200 

.7278 

1.37 

.2650 

.3211 

1.87 

.5007 

.5213 

2.37 

.7244 

.7320 

1.38 

.2699 

.3249 

1.88 

.5052 

.5254 

2.38 

.7289 

.7363 

1.39 

.2748 

.3288 

1.89 

.5098 

.5296 

2.38 

.7333 

.7406 

1.40 

.2797 

.3326 

1.90 

.5143 

.5337 

2.40 

.7377 

.7448 

1.41 

.2846 

.3365 

1.91 

.5188 

.5379 

1  2.41 

.7421 

.7491 

1.42 

.2895 

.3403 

1.92 

.5234 

.5421 

2.42 

.7465 

.7534 

1.43 

.2944 

.3442 

1.93 

.5279 

.5462 

2.43 

.7509 

.7577 

1.44- 

.2993 

.3481 

1.94 

.5324 

.5504 

2.44 

.75.53 

.7619 

1.45 

.3041 

.3520 

1.95 

.5370 

.5545 

2.45 

.7597 

.7662 

1.46 

.3090 

.3559 

1.96 

.5415 

.5687 

2.46 

.7642 

.7705 

1.47 

.3138 

.3598 

1.97 

.5460 

.5629 

2.47 

.7686 

.7748 

1.48 

.3186 

.3637 

1.98 

.5505 

.5671 

1  2.48 

.7730 

.7791 

1.49 

.3234 

.3676 

1.99 

.5550 

.5713 

2.49 

.7774 

.7833 

1.50 

0.3282 

0.3715 

[  2.00 

0.5595 

0.5754 

i  2-50 

0.7818 

0.7876 

APPENDIX. 

TABLE    III.  —  Hyperbolic  Functions 


!85 


.T 

I  sinhx 

I  cosh  X 

X 

I  sinh  X 

Z  cosh  a; 

X 

I  sinh  X 

I  cosh  X 

'  2.50 

0.7S1S 

0.7876 

2.75 

0.8915 

0.8951 

3.0 

1.0008 

1.0029 

2.51 

.7S62 

.7919 

2.76 

.8959 

.8994 

3.1 

1.0444 

1.0462 

2.52 

.7906 

.7962 

2.77 

.9003 

.9037 

3.2 

1.0880 

1.0894 

2.53 

.7950 

.8005 

2.78 

.9046 

.9080 

3.3 

1.1316 

1.1327 

2.54 

.7994 

.8048 

2.79 

.9090 

.9123 

3.4 

1.1751 

1.1761 

2.55 

.8038 

.8091 

2.80 

.9134 

.9166 

3.5 

1.2186 

1.2194 

j  2.56 

.8082 

.8134 

2.81 

.9178 

.9209 

3.6 

1.2621 

1.2628 

2.57 

.8126 

.8176 

2.82 

.9221 

.9252 

3.7 

1.3056 

1.3061 

2.5S 

.8169 

.8219 

2.83 

.9265 

.9295 

3.8 

1.3491 

1.3495 

1  2.59 

.8213 

.8262 

2.84 

.9309 

.9338 

3.9 

1.3925 

1.3929 

2.60 

.8257 

.8305 

2.85 

.9353 

.9382 

4.0 

1.4360 

1.4363 

2.61 

.8301 

.8348 

2.86 

.9396 

.9425 

4.1 

1.4795 

1.4797 

2.62 

.8345 

.8391 

2.87 

.9440 

.9468 

4.2 

1.5229 

1.5231 

2.63 

.8389 

.8434 

2.88 

.9484 

.9511 

4.3 

1.5664 

1.5665 

2.64 

.8433 

.8477 

2.89 

.9527 

.9554 

4.4 

1.6098 

1.6099 

2.65 

.8477 

.8520 

2  90 

.9571 

.9597 

4.5 

1.6532 

1.6533 

1  2.66 

.8521 

.8563 

2.91 

.9615 

.9641 

4.6 

1.6967 

1.6968 

i  2.67 

.8564 

.8606 

2.92 

.9658 

.9684 

4.7 

1.7401 

1.7402 

2.6S 

.8608 

.8649 

2.93 

.9702 

.9727 

4.8 

1.7836 

1.7836 

2.69 

.8652 

.8692 

2.94 

.9746 

.9770 

4.9 

1.8270 

1.8270 

2.70 

.8696 

.8735 

2.95 

.9789 

.9813 

5.0 

1.8704 

1.8705 

2.71 

.8740 

.8778 

2.96 

.9833 

.9856 

6.0 

2.3047 

2.3047 

2.72 

.8784 

.8821 

2.97 

.9877 

.9900 

7.0 

2.7390 

2.7390 

2.73 

.8827 

.8864 

2.98 

.9920 

.9943 

8.0 

3.1733 

3.1733 

2.74 

.8871 

.8907 

2.99 

.9964 

.9986 

9.0 

3.6076 

3.6076 

2.75 

0.8915 

0.8951 

3.00 

l.OOOS 

1.0029 

10.0 

4.0419 

4.0419 

286  APPENDIX. 

TABLE    IV. — Roots  of  Bessel's  Functions. 


-forJo(x)  =  0 

-forJ"i(x)=0 

-  for  Jo{x)  =  0 

-for  Ji(x)=0 

1 

2 
3 
4 

s 

6 

0.7655 
1.7571 
2.7546 
3.7534 

4.7527 
5.7522 

1.2197 
2.2330 
3.23S3 
4.2411 
5.2428 
6.2439 

7 
8 
9 
10 
11 
12 

6.7519 
7.7516 
8.7514 
9.7513 
10.7512 
11.7511 

7.244S 
8.2454 
9.2459 
10.2463 
11.2466 
12.2469 

TABLE   v.— Boots 

OF  J„(x)  = 

-.0. 

71  =  0 

n  =  l 

n=2 

n  —  S 

71  =  4 

71  =  5 

1 

2.405 

3.832 

5.135 

6.379 

7.586 

8.780 

2 

5.520 

7.016 

8.417 

9.760 

11.064 

12.339 

8.654 

10.173 

11.620 

13.017 

14.373 

15.700 

4 

11.792 

13.323 

14.796 

16.224 

17.616 

18.982 

5 

14.931 

16.470 

17.960 

19.410 

20.827 

22.220 

6 

18.071 

19.616 

21.117 

22.583 

24.018 

25.431 

7 

21.212 

22.760 

24.270 

25.749 

27.200 

28.628 

8 

24.353 

25.903 

27.421 

28.909 

30.371 

31.813 

9 

27.494 

29.047 

30.571 

32.050 

33.512 

34.983 

APPENDIX. 


287 


TABLE    VI.  —  Bessel's  Functions. 


X 

/o(x) 

Mx) 

X 

Mx) 

1 
J'i(x) 

X 

J,{x) 

Ji(x) 

0.0 

1.0000 

0.0000 

5.0 

—.1776 

—.3276 

10.0 

—.2459 

.0435 

0.1 

.9975 

.0499 

5.1 

—.1443 

-.3371 

10.1 

—.2490 

.0184 

0.2 

.9900 

.0995 

5.2 

—.1103 

—.3432 

10.2 

—.2496 

—.0066 

0.3 

.9776 

.1483 

5.3 

—.0758 

—.3460 

10.3 

—.2477 

-.0313 

0.4 

.9604 

.1960 

5.4 

—.0412 

-.3453 

10.4 

—.2434 

-.0555 

!   0.5 

.9385 

.2423 

5.5 

-.0068 

-.3414 

10.5 

—.2366 

—.0789 

0.6 

.9120 

.2867 

5.6 

.0270 

-.3343 

10.6 

-.2276 

—.1012 

0.7 

.8812 

.3290 

5.7 

.0599 

-.3241 

10.7 

-.2164 

—.1224 

o.s 

.8463 

.3688 

5.8 

.0917 

-.3110 

10.8 

-.2032 

—.1422 

0.9 

.8075 

.4060 

5.9 

.1220 

-.2951 

10.9 

-.1881 

—.1604 

1.0 

.7652 

.4401 

6.0 

.1506 

—.2767 

11.0 

—.1712 

-.1768 

'      1.1 

.7196 

.4709 

6.1 

.1773 

—.2559 

11.1 

—.1528 

-.1913 

!   1.2 

.6711 

.4983 

6.2 

.2017 

—.2329 

11.2 

—.1330 

-.2039 

1   1.3 

.6201 

.5220 

6.3 

.2238 

-.2081 

11.3 

—.1121 

-.2143 

1   1.4 

.5669 

.5419 

6.4 

.2433 

—.1816 

11.4 

—.0902 

—.2225 

1 

'   1.5 

.5118 

.5579 

6.5 

.2601 

-.1538 

11.5 

—.0677 

—.2284 

1.6 

.4554 

.5699 

6.6 

.2740 

—.1250 

11.6 

—.0446 

—.2320 

1.7 

.3980 

.5778 

6.7 

.2851 

—.0953 

11.7 

—.0213 

—.2333 

l.S 

.3400 

.5815 

6.8 

.2931 

—.0652 

11.8 

.0020 

—.2323 

1.9 

.2818 

.5812 

6.9 

.2981 

—.0349 

11.9 

.0250 

—.2290 

2.0 

.2239 

.5767 

7.0 

.3001 

-.0047 

12.0 

.0477 

—.2234 

1   2.1 

.1666 

.5683 

7.1 

.2991 

.0252 

12.1 

.0697 

—.2157 

■   2.2 

.1104 

.5560  . 

7.2 

.2951 

.0543 

12.2 

.0908 

—.2060 

2.3 

.0555 

.5399 

7.3 

.2882 

.0826 

12.3 

.1108 

—.1943 

2.4 

.0025 

.5202 

7.4 

.2786 

.1096 

12.4 

.1296 

—.1807 

2.5 

—.0484 

.4971 

7.5 

.2663 

.1352 

12.5 

.1469 

-.1655 

'   2.6 

—.0968 

.4708 

7.6 

.2516 

.1592 

12.6 

.1626 

-.1487 

2.7 

—.1424 

.4416 

7.7 

.2346 

.1813 

12.7 

.1766 

—.1307 

2.8 

—.1850 

.4097 

7.8 

.2154 

.2014 

12.8 

.1887 

-.1114 

2.9 

—.2243 

.3754 

7.9 

.1944 

.2192 

12.9 

.1988 

—.0912 

3.0 

—.2601 

.3391 

8.0 

.1717 

.2346 

13.0 

.2069 

—.0703 

3.1 

—.2921 

.3009 

8.1 

.1475 

.2476 

i  13.1 

.2129 

—.0489 

3.2 

—.3202 

.2613 

8.2 

.1222 

.2580 

1  13.2 

.2167 

—.0271 

?,.?, 

—.3443 

.2207 

8.3 

.0960 

.2657 

1  13.3 

.2183 

—.0052 

3.4 

—.3643 

.1792 

8.4 

.0692 

.2708 

13.4 

.2177 

.0166 

3.5 

-.3801 

.1374 

8.5 

.0419 

.2731 

13.5 

.2150 

.0380 

3.6 

—.3918 

.0955 

S.6 

.0146 

.2728 

13.6 

.2101 

.0590 

3.7 

—.3992 

.0538 

8.7 

-.0125 

.2697 

13.7 

.2032 

.0791 

3.S 

—.4026 

.0128 

8.8 

—.0392 

.2641 

13.8 

.1943 

.0984 

3.9 

—.4018 

-.0272 

8.9 

—.0653 

.2559 

13.9 

.1836 

.1166 

4.0 

—.3972 

—.0660 

9.0 

—.0903 

.2453 

14.0 

.1711 

.1334 

1   4.1 

—.3887 

—.1033 

9.1 

—.1142 

.2324 

14.1 

.1570 

.1488 

4.2 

—.3766 

—.1386 

9.2 

—.1367 

.2174 

14.2 

.1414 

.1626 

1   4.3 

—.3610 

—.1719 

9.3 

—.1577 

.2004 

14.3 

.1245 

.1747 

1   4.4 

-.3423 

-.2028 

9.4 

—.1768 

.1816 

14.4 

.1065 

.1850 

4.5 

—.3205 

-.2311 

9.5 

—.1939 

.1613 

14.5 

.0875 

.1934 

i   4.6 

—.2961 

-.2566 

9.6 

—.2090 

.1395 

14.6 

.0679 

.1999 

4.7 

—.2693 

—.2791 

9.7 

—.2218 

.1166 

14.7 

.0476 

.2043 

4.8 

—.2404 

-.2985 

9.8 

—.2323 

.0928 

14.8 

.0271 

.2066 

4.9 

—.2097 

-.3147 

9.9 

—.2403 

.0684 

14.9 

.0064 

.2069 

1   5.0 

1 

—.1776 

—.3276 

10.0 

—.2459 

.0435 

15.0 

—.0142 

.2051 

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